"This week alone, since the Russian pseudo-referendum, dozens of population centres have been liberated. These are in Kherson, Kharkiv, Luhansk and Donetsk regions all together," he said in a Tuesday nighttime address.

Last month, referendums were held on joining Russia in four regions, and Moscow used the overwhelming vote in favour as grounds for annexation of the Donetsk, Luhansk, Kherson and Zaporizhzhia regions. The vote was widely condemned by the West as rigged.

Zelenskiy cited eight small towns in Kherson in the south by name as recently having been recaptured. Reuters could not independently verify his statements.

A video released by the Ukraine defence ministry appeared to show the Ukrainian flag being raised over one of those communities, Davydiv Brid, in Kherson.

Ukrainian forces retook several villages in an advance along the strategic Dnipro River on Monday, Ukrainian officials and a Russian-backed leader in the area said.

In the east, Ukrainian forces have been expanding an offensive after capturing the main Russian bastion in the north of Donetsk, the town of Lyman, hours after Putin proclaimed the annexation of the province last week.

Russian forces in the Donetsk and Kherson regions have been forced to retreat in recent days and appear to be struggling to halt an increasingly Western-equipped Ukrainian army.

Russian President Vladimir Putin had been expected on Tuesday evening to sign a law formally annexing the four Ukrainian regions. They represent about 18 per cent of Ukraine's territory, and Kyiv and its Western allies annexation say is illegal and will not be recognised.

Russia does not fully control any of the four regions it claims - Donetsk and Luhansk in eastern Ukraine and Zaporizhzhia and Kherson in the south - and the Kremlin has said it has yet to determine the final borders of the annexed territory.

Russia has escalated its seven-month war with the annexation drive, a military mobilisation and warnings of a possible recourse to nuclear weapons to protect all of its territory.

Moscow hopes a "partial mobilisation" it announced two weeks ago can help reverse a series of battlefield setbacks.

Defence Minister Sergei Shoigu was cited by the RIA news agency on Tuesday as saying that Russia had so far called up more than 200,000 reservists out of a planned 300,000 men.

Russian defence ministry maps presented on Tuesday appeared to show rapid withdrawals of Russian forces from areas in eastern and southern Ukraine where they have been under severe pressure from the Ukrainian counteroffensive.

The ministry's daily video briefing made no mention of any pullbacks, but on maps used to show the location of purported Russian strikes, the shaded area designating Russian military control was much smaller than the day before.

On the eastern front, Denis Pushilin, the Russia-backed leader in Donetsk, said Russian forces were building a serious line of defence around the city of Kreminna after being pushed back.

In the early hours of Tuesday morning, a missile crashed into the outskirts of the Ukraine-controlled eastern city of Kramatorsk. A Reuters reporter on the scene said the missile had gouged a huge crater in the backyard of a house.

US President Joe Biden told Zelenskiy in a call on Tuesday that Washington would provide Kyiv with $US625 million ($A961 million) in new security assistance, including High Mobility Artillery Rocket System (HIMARS) launchers.

The 193-member UN General Assembly will meet on Monday over Russia's annexation. The General Assembly is expected to vote next week on a draft resolution denouncing Russia's action, diplomats said. Russia vetoed a similar resolution in the 15-member Security Council last week.

In a decree on Tuesday, Zelenskiy formally declared any talks with Putin "impossible", while leaving the door open to talks with Moscow if it got a new leader.

Australian Associated Press