Thousands of Russian troops backed by artillery and rocket barrages were advancing in what Ukrainian officials have called the Battle of the Donbas.
Russia's nearly eight-week-long invasion has failed to capture any of Ukraine's largest cities, forcing Moscow to refocus in and around separatist regions.
The biggest attack on a European state since 1945 has, however, seen nearly 5 million people flee abroad and reduced cities to rubble.
Russia was hitting the Azovstal steel plant, the main remaining stronghold in Mariupol, with bunker-buster bombs, a Ukrainian presidential adviser said late on Tuesday. Reuters could not verify the details.
"The world watches the murder of children online and remains silent," adviser Mykhailo Podolyak wrote on Twitter.
After an earlier ultimatum to surrender lapsed and as midnight approached, Russia's defence ministry said not a single Ukrainian soldier had laid down their weapons and it renewed the proposal. Ukrainian commanders have vowed not to surrender.
"Russia's armed forces, based purely on humanitarian principles, again propose that the fighters of nationalist battalions and foreign mercenaries cease their military operations from 1400 Moscow time on 20th April and lay down arms," the Russian Defence Ministry said.
The United States, Canada and Britain said they would send more artillery weaponry, and the White House said new sanctions were being prepared.
U.S. President Joe Biden is expected to announce a new military aid package about the same size as last week's $800 million one in the coming days, sources told Reuters.
U.N. Secretary-General Antonio Guterres called for a four-day humanitarian pause in the fighting this coming weekend, when Orthodox Christians celebrate Easter, to allow civilians to escape and humanitarian aid to be delivered.
Russia's war in Ukraine is to blame for exacerbating "already dire" world food insecurity, with price and supply shocks adding to global inflationary pressures, U.S. Treasury Secretary Janet Yellen said.
CITY CAPTURED
Russia says it launched what it calls a "special military operation" on Feb. 24 to demilitarise and "denazify" Ukraine. Kyiv and its Western allies reject that as a false pretext.
Ukraine said the new assault had resulted in the capture of Kreminna, an administrative centre of 18,000 people in Luhansk, one of the two Donbas provinces.
Russian Foreign Minister Sergei Lavrov confirmed that "another stage of this operation is beginning".
Driven back by Ukrainian forces in March from an assault on Kyiv in the north, Russia has instead poured troops into the east for the Donbas offensive.
It has also made long-distance strikes at other targets including the capital and the northeastern city of Kharkiv, where at least four people were killed by missiles, authorities said on Tuesday.
In one suburban street, the body of an elderly man lay face down near a park, a thick ribbon of blood running into the gutter.
"He worked in security not far from here," a resident named Maksym told Reuters. "The shelling began and everyone fled. Then we came out here, the old guy was already dead."
MARIUPOL
In Mariupol, scene of the war's heaviest fighting and worst humanitarian catastrophe, about 120 civilians living next to the Azovstal steel plant left via humanitarian corridors, the Interfax news agency said on Tuesday, quoting Russian state TV.
A drone footage captured on Tuesday shows people buying food and other necessities at a makeshift market, as well as charging their mobile phones from a generator for about $1.35.
A Reuters correspondent said prices at the market were extremely high versus what people would normally pay there.
Mariupol has been besieged since the war's early days. Tens of thousands of residents have been trapped and Ukraine believes more than 20,000 civilians have died there.
"The Russian army will forever inscribe itself in world history as perhaps the most barbaric and inhuman army in the world," Ukrainian President Volodymyr Zelenskiy said.
"Deliberately killing civilians, destroying residential quarters and civilian infrastructure, and using all kinds of weapons, including those prohibited by international conventions, is already the brand signature of the Russian army," he added in a video address.
Russia has denied using banned weapons or targeting civilians in its invasion of Ukraine and says, without evidence, that signs of atrocities were staged.
Video released by Ukraine's Azov battalion purported to show people living in the underground network beneath the sprawling steel plant, where they say hundreds of women, children and elderly civilians are sheltering with diminishing supplies.
"We lost our home; we lost our livelihood. We want to live a normal, peaceful life. We want to get out of here," an unidentified woman says in the video.
"There are lots of children in here - they're hungry. Get us out of here, we beg you. We've already cried out all the tears we have. We can't cry anymore," she added.
Reuters could not independently verify where or when the video was shot.
(Reporting by Reuters journalists; Writing by Costas Pitas and Lincoln Feast; Editing by Himani Sarkar)
