Russian forces on Friday pounded the last Ukrainian strongholds in a separatist-controlled eastern province of Ukraine, including a city where authorities said 1,500 people have been killed and 60% of residential buildings destroyed since the start of the war.

Ukraine's foreign minister warned that without a new injection of foreign weapons, Ukrainian forces would not be able to stop Russia from seizing Sievierodonetsk and nearby Lysychansk, locations that are crucial to Russia's goal of capturing all of Ukraine's industrial Donbas region.

The cities are the last areas under Ukrainian control in Luhansk, one of two provinces that make up the region.

Russian forces have made slow but persistent advances as they bombarded and sought to encircle both Lysychansk and Sievierodonetsk.

"The Russians are pounding residential neighbourhoods relentlessly," regional governor Serhiy Haidai wrote in a Telegram post on Friday.

"The residents of Sievierodonetsk have forgotten when was the last time there was silence in the city for at least half an hour."

Russian shelling killed four people in the city over the past 24 hours, he said.

Mayor Oleksandr Stryuk said late on Thursday that at least 1,500 people have been killed in Sievierodonetsk since Russia invaded Ukraine on February 24.

About 12,000 to 13,000 remain in the city - down from a pre-war population of about 100,000 - and 60% of residential buildings have been destroyed, he said.

Mr Stryuk said a Russian reconnaissance and sabotage group entered a city hotel, and that the main road between neighbouring Lysychansk and the city of Bakhmut to the south-west remains open, but travel is dangerous. He said only 12 people were able to be evacuated on Thursday.

In Donetsk, the Donbas region's other province, Russia-backed rebels claimed on Friday to have taken control of Lyman, a large railway hub north of two more key cities that remained under Ukrainian control. There was no immediate confirmation from Ukrainian officials.

With Ukraine's hopes of stopping the Russian advance fading, Ukrainian Foreign Minister Dmytro Kuleba pleaded with Western nations to provide his country with more weapons so its defenders were equipped to "push (the Russian forces) back".

"We need heavy weapons. The only position where Russia is better than us, it's the amount of heavy weapons they have. Without artillery, without multiple launch rocket systems we won't be able to push them back," Mr Kuleba said in a video posted on Twitter on Thursday night.

He said the situation in the east was "even worse than people say. ... If you really care for Ukraine, weapons, weapons and weapons again."

In his nightly address to the nation, Ukrainian President Volodymyr Zelensky had some harsh words for the European Union, which has not agreed on a sixth round of sanctions that includes an embargo on Russian oil.

"Of course, I am grateful to our friends who are promoting new sanctions," the Ukrainian leader said. "But where did those who block the sixth package get so much power? Why are they still allowed to have so much power, including in intra-European procedures?"

Mr Zelensky also spoke bluntly about what is at stake in the battle for eastern Ukraine.

"Pressure on Russia is literally a matter of saving lives," he said. "And every day of delay, weakness, various disputes or proposals to 'appease' the aggressor at the expense of the victim is new killed Ukrainians. And new threats to everyone on our continent."

Moscow pressed the West on Thursday to lift sanctions already imposed over the war, seeking to shift the blame for a growing global food crisis that has been worsened by Kyiv's inability to ship millions of tons of grain and other agricultural products while under attack.

Britain immediately accused Russia of "trying to hold the world to ransom," insisting there would be no sanctions relief, and a top US diplomat blasted the "sheer barbarity, sadistic cruelty and lawlessness" of the invasion.