Curtis Sliwa, his Republican opponent, noted that Mr. Adams’s hotels-to-housing plan focuses on the outer boroughs, when most of the hotels that have closed are actually in Manhattan. “You are going to stick more homeless people in Staten Island, Brooklyn, Queens and the Bronx and exempt Manhattan?” he said. “Come on.”

The number of single adults in the city’s main shelter system has fallen slightly from a record high in the spring, but is still up about 60 percent since Mr. de Blasio took office in 2014. The family shelter population has dropped, but largely because of an eviction moratorium that is set to expire in January.

In parts of Manhattan, many random attacks and hate crimes have been linked to people living in streets and on subways with untreated mental illness. Assaults in the precinct around Times Square have more than doubled since before the pandemic, and felony assaults in the transit system are up more than 35 percent compared with 2019, even as ridership has dropped. The city’s response has included aggressively tearing down homeless encampments, a tactic advocates say merely pushes people from place to place.

Topically, the final debate before next Tuesday’s mayoral election has been similar to last week’s debate. The candidates largely agree on the future of New York City’s gifted and talented program (they would expand it). Mr. Adams supports some vaccine mandates for civil servants and students. Mr. Sliwa does not.

But tonally, tonight’s debate has proven far nastier.

After Mr. Adams argued he would have engaged more energetically with union leaders surrounding vaccine mandates, Mr. Sliwa suggested Mr. Adams talk to his “friend and teammate” Bill de Blasio, the outgoing mayor.

“You are acting like my son when he was 4 years old,” Mr. Adams shot back. “Show some discipline so we can get to all of these issues. You’re interrupting, you’re being disrespectful.”

Mr. Sliwa countered that Mr. Adams should stop being a “robot” and show some compassion for city workers who risk losing their salaries for failing to get vaccinated.

Soon enough, the conversation — or bare-knuckled brawl — got even more personal.

Mr. Sliwa accused Mr. Adams of actually living in New Jersey, an allusion to some of the cloudiness surrounding Mr. Adams’s residency, and he mocked Mr. Adams’s decision to blame his tax-filing errors on his purportedly homeless accountant.

“You fake where you live, Eric Adams,” Mr. Sliwa said.

Mr. Adams said Mr. Sliwa was demonstrating “clown-like actions,” and then accused him of hiding money so he would not have to pay child support.

“That is scurrilous that you would say that,” Mr. Sliwa said.

“I would have engaged with credible messages, and that is how you get things done,” Mr. Adams said.

In last week’s debate he ultimately said he would follow through with the mayor’s decision to put on leave or fire police officers and firefighters who defy the order — but only when pressed by the moderator to answer. “Yes, I will follow the orders that are in place,” he said.

Mr. Adams also said that he supports the city’s “Key to NYC” policy, which requires people to have received at least one dose of a vaccine to participate in indoor dining and entertainment. The mandates “have both proven to be successful in significantly boosting vaccinations in our city, which is expediting our return to normalcy,” he told The New York Times in a statement.

Curtis Sliwa, on the other hand, consistently rails against vaccine mandates, though he is vaccinated and says he wants others to be — if they so choose.

Mr. Sliwa participated in Monday’s large anti-mandate protest in which thousands of people — including many firefighters, police officers and sanitation workers — marched across the Brooklyn Bridge and demonstrated in front of City Hall. At the debate Tuesday, he called the policy, which will require unvaccinated workers to go on unpaid leave as of Nov. 1, “madness.”

“They could have been tested once a week if they cannot be vaccinated or will not be vaccinated,” he said. “It was working with the police, it was working with the firefighters.”

Mr. Sliwa also said at the debate that it was “ridiculous” that he has to show a vaccine card and I.D. to “get a cheeseburger and fries” in New York City, using question about outdoor dining to pivot to a critique of the vaccine mandate for people seeking to dine indoors.

His feline collection began six years ago when he moved in with his fourth wife, Nancy Sliwa, in her apartment near Central Park. They took in rescue cats that were sick or abandoned.

Now Mr. Sliwa is making animal welfare a central part of his campaign. He released a “13-Point Animal Welfare Plan” last week that includes creating a “no-kill” shelter system and ending the horse carriage industry.

His first television ad featured him holding one of his cats, Tuna, and promising “compassionate solutions” as mayor.

During a reporter’s visit to his home over the summer, the cats climbed onto the dining table, walked across a photographer’s lap and gathered in a front window to watch pigeons. The apartment did not smell bad.

“You change the litter three times a day,” Mr. Sliwa said.

On homelessness, Mr. Adams has said that he carried a garbage bag full of clothes to school as a child because he was worried that his family would be evicted before he returned home.

On crime and public safety, Mr. Adams promises that he can both promote public safety and protect Black and Latino residents from civil rights abuses because he was beaten by the police as a teenager, but then joined the Police Department and spoke out against discriminatory behavior from within its ranks.

Since he won the Democratic primary, Mr. Adams has held a raft of fund-raisers with New York’s rich and powerful. He’s consulted with the billionaire former mayor Michael R. Bloomberg and dined with the billionaire media mogul Rupert Murdoch while promising that New York City government will be more friendly to business.

This ability to be many things to many people and to convincingly shape-shift at will is Mr. Adams’s greatest skill but may also be his greatest liability. By appealing to so many varied constituencies at once, Mr. Adams has created a tent so large that disappointment is inevitable.

Even those who have known Mr. Adams for decades aren’t sure which version will show up to City Hall if, as expected, he wins the Nov. 2 election and becomes the city’s second Black mayor. Sometimes, even Mr. Adams does not seem to be sure.

“I’m so many formers,” Mr. Adams said during a July visit to the White House, where he declared himself the new face of the Democratic Party, “I’m trying to figure out the current.”

“Eric has been wined, dined, and pocket-lined by the uber-rich, realtors, developers, and hedge fund monsters,” Mr. Sliwa said in a recent news release accompanying a video of himself trying to enter a private club where he said Mr. Adams was socializing. “Elite Eric has been bossed and bought.”

Mr. Sliwa, the founder of the Guardian Angels patrol group and a popular AM radio personality, lives in a 320-square-foot apartment on the Upper West Side with his fourth wife and more than a dozen cats. In recent weeks, he has mocked Mr. Adams’s tailored suits and his predilection for frequenting an exclusive private club in NoHo, Zero Bond.

Mr. Adams, who grew up in poverty before becoming a police captain, state senator and Brooklyn borough president, has in fact displayed a taste for New York City’s exclusive nightlife. He has made repeat appearances in the New York Post’s Page Six, after spending multiple nights at Zero Bond, whose patrons include Paris Hilton, and at Rao’s, the famously exclusive Italian joint in East Harlem.

After Mr. Adams went on a post-primary vacation to a European country he declined to name, Politico reported the destination was Monaco, a principality known for its popularity with the idle rich.

“Who goes to Monaco?” Mr. Sliwa asked during last week’s debate.

Mr. Adams has won the support of several billionaires, who have donated to him directly and funded a super PAC that campaigned on his behalf in the Democratic primary.

In recent days, Mr. Sliwa has also cast himself as the pro-motorist candidate, while Mr. Adams recently won the endorsement of StreetsPAC, a group that wants to rein in drivers on city streets.

The moderators had threatened to cut off the microphone if the rules weren’t obeyed, but it often took at least two moderators to get Mr. Sliwa to stop speaking. And while Mr. Adams kept his composure, he complained more than once that the moderators were not enforcing the rules.

The task of keeping things in line tonight will fall to three respected veterans of New York City journalism: Bill Ritter, a WABC-TV anchor; Dave Evans, a WABC-TV political reporter; and Mariela Salgado, a news anchor from Univision 41.

Mr. Ritter, a native of Los Angeles who has worked as a journalist since 1972, will serve as the moderator. He started working at ABC in 1992 as a founding co-host of “Good Morning America Sunday.” Mr. Ritter was kicked out of San Diego State University for protesting against the Vietnam War, but returned to college 40 years later and received his degree at the New School in 2016. He has served as the moderator on mayoral debates airing on the station since 2001.

Mr. Evans, a panelist, arrived at WABC-TV in 1999 and has covered everything from the war in Iraq to the 2008 presidential campaign of Hillary Clinton. He has competed in many marathons and triathlons.

Ms. Salgado, also a panelist, is a native of Chile who has been a journalist for more than 20 years. She has reported on the disputed ballots in Florida during the 2000 election, the Sept. 11 terror attacks, and the plight of undocumented Central Americans as they made the dangerous journey to the United States.

But Mr. Sliwa may try to seize any opportunity he can to stoke doubts about Eric Adams, the Democratic nominee. On the debate stage last week, the exchanges were relatively mild, and Mr. Adams ignored many of Mr. Sliwa’s criticisms.

For Mr. Adams, the task is the opposite: Do nothing that would change the dynamics of the race, while seeking to fuel enthusiasm in an election in which turnout is unpredictable and many voters are fatigued by politics.

Can he stay above the fray, as he did last week, or will he find himself drawn into a tit-for-tat with Mr. Sliwa? Can he offer a statesmanlike vision for the city while energizing viewers to turn out, or will he be put onto the defensive by moderators or his opponent?

Many political observers expect that the debate will do little to change the direction of the election, but it could affect how voters feel about their choices.

Bill Ritter, a WABC-TV anchor who has moderated mayoral debates for almost two decades, will serve as moderator. He’ll be joined by two panelists: Dave Evans, the station’s longtime political reporter, and Mariela Salgado, an anchor for Univision 41.

A team of reporters from The New York Times will be offering live commentary and analysis as the debate unfolds.

Mr. Sliwa sought to tie Mr. Adams to Mayor Bill de Blasio, who is unpopular in many corners of the city, while arguing that Mr. Adams was out of touch with working class New Yorkers. Mr. Sliwa has kept up those attacks in recent days.

“Eric Adams loves New York City so much that he lives in an apartment in New Jersey, spends his summers in Monaco, and hosts his parties out in the Hamptons,” Mr. Sliwa wrote on Twitter on Sunday, raising questions about Mr. Adams’s residency, a recent vacation to Europe and his fund-raisers on Long Island.

The debate also covered a broad array of critical issues facing New York City, from the crisis at the Rikers Island jail facility to a new vaccine mandate for city workers. Mr. Adams wants to close Rikers; Mr. Sliwa vowed to move there temporarily in his first days in office. Mr. Adams supports the vaccine mandate; Mr. Sliwa does not.

Mr. Adams agrees with a decision to remove a Thomas Jefferson statue from City Council chambers; Mr. Sliwa wants to keep the statue there.

Over the course of the hourlong debate, Mr. Adams tried to remain calm and argued that Mr. Sliwa’s confession that he made up crimes for publicity in the 1980s was disqualifying.

“He made up crime, New Yorkers,” Mr. Adams said. “That in itself is a crime.”

His Republican opponent, Curtis Sliwa, will be trying to increase his odds on Tuesday, taking one last shot at knocking Mr. Adams off his glide path to City Hall. But so far, Mr. Sliwa, the flamboyant founder of the Guardian Angels patrol group and longtime media personality, has failed to break through to voters with a populist platform focused on public safety, animal rights and the plight of New York City’s homeless population.

The first Adams-Sliwa debate last week did little to shake up the race’s dynamic, even as the candidates disagreed on vaccine mandates for students, how to fight a rise in crime across the city and a congestion pricing plan.

Mr. Adams stuck to broad platitudes, promising to make New York City safer and more prosperous. He largely ignored the bait flung by Mr. Sliwa, who tried to lure him into fights over whether he actually lived in New York City and where he vacations.

“I’m speaking to New Yorkers,” Mr. Adams quipped. “Not speaking to buffoonery.”

But Mr. Adams also eluded questions from moderators eager to pin down more specific policy plans from a candidate whose views have repeatedly evolved.

Rallying with prominent Democrats on Sunday, Mr. Adams shared a piece of campaign advice for himself in the race’s final week: “You don’t win a baseball game in the eighth inning.”

Voters began casting ballots early across the city in modest numbers last weekend and can do so every day through Sunday.

On stop-and-frisk, which at its peak involved the stops of hundreds of thousands of people who were often young Black and Latino men for weapons that rarely materialized, Mr. Sliwa, the Republican nominee, attempted to suggest that Mr. Adams would support the abuse of the practice.

“I would not do what one of his main supporters, Michael Bloomberg, did,” Mr. Sliwa said, “which has to have 700,000 predominately men of color be stopped and frisked with a quote system that was put on.”

Mr. Adams, a former police captain and the Democratic nominee, shot back that his record as an officer and politician showed he had “protected Black and brown and low-income New Yorkers” from police abuses.

“My son was a victim of stop-and-frisk in the city,” Mr. Adams replied. “I never call for aggressive police tactics, I call for appropriate police tactics.”

After last summer’s protests over the murder of George Floyd, public pressure to shrink police budgets and rein in the department’s mission has mounted. At the same time, the number of shootings and murders in the city rose last summer to their highest levels in at least a decade and have remained well above prepandemic levels ever since.

The department is also dealing with a crisis of public confidence. Documents released over the last year have shown how abusive officers almost always remain on the force, often receiving little to no punishment, even when instances of police brutality have been substantiated by department or city investigators.

In one clash, Mr. Sliwa criticized Mr. Adams for not interviewing Hispanic and Latino candidates to be the next police commissioner. Mr. Adams, who has said he would select a woman of color for the role, replied that Mr. Sliwa did not know who he had interviewed and told him to focus on his own plans.

Mr. Adams attempted to strike a stance he took throughout the primary season, detailing a measured approach to policing overall, while emphasizing public safety as a core campaign issue. He has said he believes in responsibly trimming fat from the police department’s budget.

“I’ve been clear on this message for the last 35 years,” Mr. Adams said. “I have not changed at all.”

But Mr. Sliwa, who has also emphasized a commitment to public safety, criticized Mr. Adams for his position on the budget, saying he not only believes the funding should be kept intact, but that he believes it should grow. And he attacked Mr. Adams’s previous remarks that he would carry a gun as mayor.

“I’m in the inner-city all the time, I don’t wear a bulletproof vest, I don’t carry a gun. I never have,” Mr. Sliwa said. “I’ll tell you, if you’re going to reach young men who are using guns in violent actions, you cannot say ‘Do as I say but not as I do.”

In the debate, though, the candidates clashed over their approaches when asked to name three things they would do to prevent more flooding deaths. Mr. Adams focused on better warning systems and clearer plans for measures to take during major cloudbursts.

“We’re going to institute a clear response plan,” he said. “We learn from terrorism, you don’t wait ‘til the day when a plane attacks.”

Mr. Sliwa called for more sea walls to be built, noting that one planned in Staten Island remained unbuilt nine years after Hurricane Sandy’s coastal flooding devastated the area. “People have no faith in politicians like Eric Adams or Bill de Blasio or others,” he said.

Mr. Adams — who unlike Mr. Sliwa has made sure to frame the flooding problem as part of the larger dangers of climate change — noted correctly that the recent flooding was from downpours, not storm surges from the sea.

“It had nothing to do with sea walls,” he said, declaring that applying antiquated methods to a modern problem won’t work: “This must be intervention and prevention.” (Mr. Adams does not oppose sea walls, but he sees them as one of many tools that must be applied differently according to the needs and ecology of each area.)

While Mr. Sliwa offers a four-point policy plan that does not mention the changing climate, Mr. Adams has a plan that promises to address climate challenges, including extreme heat, sea rise and intense storms, in every neighborhood. He vows to speed the implementation of plans begun by the Bloomberg and de Blasio administrations, like expanding green roofs and wetlands restoration.

Mr. Adams has also talked about addressing the city’s contribution to climate change by reducing fossil fuel burning and phasing out gas-fired power plants, though he has not specified a timeline. Nor has he said whether he will make it a priority to implement and expand a city law requiring large buildings to reduce emissions of planet-warming gases. Real-estate groups, which backed his campaign, are seeking to limit the law and maximize loopholes.

Mr. Adams said, as he has in the past, that he takes responsibility for omissions on his tax returns, and then blamed his accountant, who he said was homeless.

“He went through real trauma,” Mr. Adams said of his accountant. “And I’m not a hypocrite, I wanted to still give him the support that he needed.”

Mr. Adams also insisted, again, that his primary residence is in Brooklyn.

Mr. Adams owns a multi-unit townhouse in the Bedford-Stuyvesant area of Brooklyn in which he says he keeps an apartment. In one of the more bizarre moments of the mayoral primary, he gave a media tour of that apartment, with reporters observing non-vegan food items apparently belonging to Mr. Adams’s son. (Mr. Adams has been a vegan for years.)

But Mr. Adams also co-owns a co-op in Fort Lee, N.J., with his partner, and he has said that he moved into Brooklyn Borough Hall for a time after the pandemic arrived. During the primary, Politico New York reported that Mr. Adams used conflicting addresses in public records and that he was still spending nights at Borough Hall.

He has had to refile his tax returns in part because of irregularities concerning his residency, the news outlet The City reported. The outlet also reported that the city is seeking to inspect his Brooklyn residence following an allegation of an illegal apartment conversion on the property. His campaign has said he intended to rectify those issues, though the complaint remains active.

Mr. Sliwa recently led a journey from Manhattan to Fort Lee “to find out where Eric Adams really lives.”

The high-stakes test, which is typically administered in January, has helped create a cottage industry of test preparation for young children in New York City.

The city’s advisory school board rejected the contract for that exam earlier this year, leaving Mr. de Blasio, who has sharply criticized the test, without a clear admissions system.

Both Mr. Adams and Mr. Sliwa have said they would keep the gifted program in place, but they have not provided details on whether they will continue administering the test, overhaul it or replace it with some other form of screening.

During the debate on Wednesday, Mr. Adams said that the city should re-examine the admissions exam, while making sure to expand opportunities for “accelerated learners, make sure it is in every district in our city and every ZIP code.”

“I made it clear that we need to look at that exam,” he said. “I don’t believe a four-year-old taking the exam should determine the rest of their school experience. That is unacceptable.”

Mr. Sliwa also reiterated his support for expanding the gifted program to all schools, noting that his son was one of thousands of students who took the test and “lost out.”

“We seem to be taking it out on Asian families and Southeast Asian families because they’re doing so well,” Mr. Sliwa said. “We need to expand gifted and talented so it is in all schools, even if only three or two children qualify.”

Mr. Sliwa will likely use the debate to highlight his central campaign themes, touching upon issues including public safety, animal welfare and confronting homelessness.

The debate presents a chance for Mr. Sliwa to surprise viewers with a sober-minded demeanor that matches the mood of an anxious city. But he is more likely to aggressively challenge the Democratic nominee, Eric Adams, on a variety of subjects, including questions of his residency.

Mr. Sliwa can also be a brawler onstage, as seen in his first major debate in the Republican primary against Fernando Mateo, where the men traded insults and continued to yell at each other, including while muted.

But even some political strategists were barely aware that the debate was occurring, suggesting that Mr. Sliwa will be able to do little to move the dial much with the general public.

Still, it has been months since Mr. Adams found himself at the center of a debate-stage clash, and many Democrats believe he would be wise to stay above the fray no matter how much Curtis Sliwa, his Republican rival, seeks to provoke him.

Instead, the debate offers a chance for Mr. Adams to lay out his vision and emphasize unity before a city that last saw him in the midst of a crowded and contentious Democratic primary battle.

Mr. Adams is best known for his public safety plans, but the debate may also challenge him to offer details about how he would handle the many other problems New York City confronts.

More broadly, it will be a moment for Mr. Adams to show that he understands the mood of a city that is by turns optimistic about continued reopening, and deeply anxious about the lingering pandemic and its attendant health, educational and economic consequences.

Sally Goldenberg, the tireless City Hall bureau chief for Politico New York, got her first job in journalism in 2002, at New Jersey’s weekly Hillsborough Beacon, where she did the police blotter and covered school board meetings. Ms. Goldenberg recently broke the news that Mr. Adams — who had declined to say where he had gone on his post-primary European vacation — had vacationed in Monaco. She enjoys the Olympics and old-school R&B.

Melissa Russo, WNBC’s political and government affairs reporter, has covered four mayors and is known for her tough investigations of issues affecting children, which have focused on problems at the New York City Housing Authority and the Administration for Children’s Services. She likes photography and a capella music.

Allan Villafaña, an anchor for Telemundo 47, was born in Puerto Rico, studied journalism at Ohio State University and spent time in Miami, where he worked as an anchor for Mega TV. An animal lover, he recently brought two dogs — Petunia Marie and Yoyo Jesus — to the blessing of the animals at Our Lady of Good Counsel Church in New Jersey on the feast day of St. Francis of Assisi.

But Mr. Sliwa has been a celebrity in New York City for decades.

He became famous in the 1980s for leading the Guardian Angels, a civilian crime-fighting group. He became a conservative radio host known for saying outrageous things. He survived a shooting that left him with five bullet wounds and testified at a federal trial against John A. Gotti, the Gambino crime family scion.

He got involved in politics, and led the Reform Party of New York State; in 2018, the last statewide election, the Reform Party drew the fewest votes for governor among 10 parties on the ballot. He became a Republican last year and decided to run for mayor, winning a bitter primary in June against his longtime friend Fernando Mateo, a restaurateur.

Mr. Sliwa has focused on a law-and-order message, and he has criticized former President Donald J. Trump and said that he did not vote for Mr. Trump in 2016 or 2020.

He acknowledges that there have been few parallels to his campaign.

“Who at the age of 67 is running around wearing a red beret and a red satin jacket and going out there like a crime fighter and a superhero from our days reading comic books?” Mr. Sliwa told The New York Times over the summer. “That’s a bit eccentric.”

If elected, Mr. Adams would be only the city’s second Black mayor.

After the June primary election, Mr. Adams, the Brooklyn borough president and a centrist, proclaimed that he was “the face of the new Democratic Party,” and vowed to “show America how to run a city.”

He has since worked to significantly expand his base, courting the city’s business community, holding several fund-raisers and consulting with former Mayor Michael R. Bloomberg and his team about how to govern.

Mr. Adams’s primary campaign focused on crime. Although the issue remains a significant focus for him, he has also begun to lay out some of his other ideas, calling for neighborhoods such as SoHo in Manhattan to be rezoned to create more affordable housing; pledging to build hundreds of miles of new bike lanes; and promising to keep and expand the city’s gifted and talented program in public schools after Mayor Bill de Blasio announced that he was phasing it out.

But for at least one hour, Mr. Adams will be forced to devote some attention to his Republican opponent, Curtis Sliwa, as they go head-to-head on Wednesday at 7 p.m. in the first of two official debates among the two leading candidates for mayor of New York City.

The one-hour debate will be aired on WNBC-TV Channel 4 and also on Telemundo, Channel 47, in Spanish.

NYC Life TV will offer a simulcast on Channel 25.1.

The debate will also be livestreamed on NBCNewYork.com, Telemundo47.com and Politico New York.

A team of reporters from The New York Times will provide live commentary and analysis for the debate, which will be moderated by four journalists: David Ushery, a news anchor for WNBC-TV in New York; Sally Goldenberg, the City Hall bureau chief for Politico New York; Melissa Russo, WNBC’s political and government affairs reporter; and Allan Villafaña, an anchor for Telemundo 47.

His Republican opponent, Curtis Sliwa, the founder of the Guardian Angels subway patrol group, has struggled to gain momentum as a candidate, and is expected to use the debates to attack Mr. Adams, who has largely ignored him on the campaign trail.

The candidates have already traded barbs in recent weeks ahead of the Nov. 2 election. Mr. Adams called Mr. Sliwa racist and said that he was making the campaign a circus. Mr. Sliwa has criticized Mr. Adams’s campaign against a whistle-blower police officer in the 1990s and has raised questions over where Mr. Adams lives.

Both candidates have focused on a law-and-order message. But Mr. Adams, a former police captain, has treated the election almost as an afterthought, proclaiming himself as the future of the national Democratic Party and holding meetings on his likely transition to the mayoralty.