Russia continues its bombardment of cities across Ukraine including in the early hours of Sunday, when it launched an assault on the port of Mykolaiv, killing a prominent businessman. But for Ukraine, Kherson is an important strategic objective as the largest population center occupied by the Russians and the first city to fall. As a port, it is economically important to the Ukrainians and taking it back would deny Russian forces access to the southern coast toward Odessa.

Mick Ryan, a military strategist and retired major general in the Australian army, said the offensive will force Russia to make hard decisions about keeping troops in the Donbas or moving them south to protect Kherson.

If the Ukrainians retake the city, he said, they could be in a position to threaten Russia’s main Black Sea naval base, 150 miles away, at Sevastopol.

The Ukrainian effort to retake Kherson represents a significant development in the conflict, said Gen. Ryan. “If the Ukrainians can take that back, that will be a turning point,” he said. “But we’re not at a turning point yet.”

Eliot Cohen, a military historian and strategist with the bipartisan policy research group the Center for Strategic and International Studies in Washington, said Kherson carried great symbolic importance.

“Taking back the original city that the Russians took without much effort in the beginning, would be psychologically very significant,” he said. It would be a bigger deal than either Ukraine’s recapture of Snake Island in June or the sinking of Russia’s flagship, the Moskva, in April.

Military offensives are more challenging than defensive operations. Analysts caution that Ukraine shouldn’t – and likely won’t – rush into the fight in the south because it must continue to check Russian advances in the east. But demonstrating that it can retake ground in the south would provide an important victory for Ukrainian morale and show its backers, particularly those in Europe as the continent faces a tough winter with likely energy shortages, that their support is yielding results on the ground.

If Ukraine’s push to dislodge Russians from Kherson fails or falters, however, it could weaken support for Kyiv’s fight in some Western capitals. Ukrainians are likely to continue fighting whatever happens, but an unsuccessful campaign could prompt more calls for a negotiated settlement, particularly from parts of Western Europe facing reduced flows of Russian natural gas.

U.S. officials say Ukrainian forces are advancing in the south, and public assessments from British defense intelligence suggest the counteroffensive in Kherson is gathering momentum. The British intelligence said Thursday that Ukrainian forces have likely established a bridgehead south of the Ingulets River, which forms the northern boundary of the Kherson region, and have damaged at least three bridges that Russia uses to deliver supplies to the area. One—the 1,100-yard Antonivsky bridge near Kherson city—is now probably unusable.

This has exposed Russia’s 49th Army, stationed on the west bank of the Dnipro River, and has cut off Kherson city from other occupied territories, the British intelligence said. On Saturday, they said Russian forces were highly likely to have established two pontoon bridges and a ferry system to compensate for the bridge damage.

This phase of the war will look different from the first one, when Moscow unsuccessfully mounted an effort to strike at Kyiv and topple the government of President

Volodymyr Zelensky,

and the second that continues in the east, where grueling exchanges of artillery fire have yielded modest advantages for Russian forces at great cost.

Mr. Cohen says this phase will likely have parallels with what happened in the last year of World War I, when the Germans on the one side and the British and Australians on the other sought to “break in” past the front lines, exploit weakness and infiltrate forces.

This requires “meticulously planned operations, which take one bite at a time out of the enemy’s front line. And then you move artillery forward, you consolidate your position, let them counterattack if they want to, and then you take another bite,” he said.

Analysts point out that this phase won’t depend on artillery alone. Konrad Muzyka, president of Rochan Consulting, military analysts based in Gdansk, Poland, said, “Himars cripple Russia’s ability to conduct offensive operations, but they won’t force the Russians to leave Ukraine. For that you need manpower and armor.”

This brings in the big unknown: “We don’t know what the structure of the Ukrainian army is, we don’t know its number of troops or the state of their morale,” he said. Ukraine has lost thousands of soldiers in recent months and many good leaders.

Chris Dougherty, a former U.S. Defense Department strategist now at the Center for a New American Security, said that, despite all the materiel the West has given to Ukraine, it probably still lacks the equipment and trained forces to retake ground successfully and quickly.

“The worry I have is we give advanced equipment to the Ukrainians and they use it to stop the bleeding,” he said. “That makes sense if you’re bleeding to death. But what’s the next thing you do?” He said Russia has been unable to capitalize on its massive artillery blitzes to take significant ground, and Ukraine risks falling into the same trap.

Mr. Dougherty said he doubts Ukrainian forces can take ground in the east, where Russian forces are well dug-in, but thinks they can do it around Kherson or other areas in the south, where operations by partisans have already been hitting Russian targets.

“They have to make sure the Russians can’t rapidly reinforce from another area,” he said. “And the Ukrainians have to make sure they take advantage of what they have – partisans and intelligence inside Kherson.”

“The Ukrainians have to find a way to hit weak points in the Russian line and hit them in the rear of their line. Nothing panics an army like knowing their line is getting hit in the rear,” he said.

Some military analysts said that given Western military resources and hardware, economic aid and critical intelligence assessments about Russian military movements, the balance in the south is shifting to the Ukrainians.

Russia has already lost tens of thousands of soldiers killed and injured in its efforts so far, and analysts suspect many units are on the brink of exhaustion. They will be looking out for indications that the Russian army is becoming brittle and signs of mutiny, desertion, combat refusal and surrenders. Meanwhile, evidence is growing that the Russian economy is hurting from Western-led sanctions.

“My view is the momentum is sort of gradually swinging to Ukrainians,” said Mr. Cohen. “In war, unpredictable things happen. People make mistakes. All that… Nothing is a certainty. But at the moment, would I rather have the Ukrainian hand to play with or the Russian hand to play? I would rather have the Ukrainian hand, provided that the West continues to pour military support and some economic support in there.”

Write to Stephen Fidler at stephen.fidler@wsj.com and Daniel Michaels at daniel.michaels@wsj.com