Russian forces were fighting on Monday to capture Lysychansk, the last major city still held by Ukrainian troops in eastern Luhansk province, after Moscow-backed separatists said they were advancing on multiple fronts.
In a victory for Kremlin's campaign, Lysychansk's twin city of Sievierodonetsk, the scene of some of the bloodiest fighting, fell to pro-Russian forces on Saturday.
Russian missiles also struck Kyiv for the first time in weeks on Sunday, attacks condemned by US President Joe Biden as "barbarism".
Tass news agency on Sunday quoted a separatist official as saying Moscow's forces had entered Lysychansk from five directions and were isolating Ukrainian defenders. However Reuters could not confirm the report.
The General Staff of Ukraine's Armed Forces said Russian forces were using artillery to try to cut off Lysychansk from the south but made no mention of separatists entering the city.
Russian assault aircraft struck near Lysychansk, the general staff added in its update on Monday.
"Lysychansk, it was a horror, the last week," said Elena, an elderly woman from the city who was among dozens of evacuees who arrived in the Ukrainian-held town of Pokrovsk by bus from frontline areas.
"I already told my husband if I die, please bury me behind the house," she added.
The RIA agency quoted a separatist official as saying separatist forces had evacuated more than 250 people, including children, on Sunday from Sievierodonetsk's Azot chemical plant.
The industrial area was the last part of Sievierodonetsk held by Ukrainian forces before they withdrew.
Luhansk and neighbouring Donetsk make up Ukraine's eastern Donbas region - the country's industrial heartland.
Ukrainian presidential adviser Oleksiy Arestovych said missiles also struck the central city of Cherkasy on Sunday, hitting a strategic bridge linking western Ukraine and the eastern battlefields.
"They are trying to limit the transfer of our reserves and Western weapons to the east," he told Reuters.
Odesa regional administration spokesperson Serhiy Bratchuk said a missile strike in the Odesa region, in southern Ukraine, had destroyed residential buildings and caused a fire. Six people were injured including a child, he added.
Ukraine needs a modern air defence system to deter Russian missiles, President Volodymyr Zelenskiy said in a Sunday evening address, after missiles earlier struck an apartment block and close to a kindergarten in the Ukrainian capital Kyiv.
"Partners need to move faster if they are really partners, not observers. Delays with the weapons transfers to our state, any restrictions - this is actually an invitation for Russia to hit again and again," he added.
The United States is likely to announce this week the purchase of an advanced medium to long range surface-to-air missile defence system for Ukraine, a source familiar with the matter told Reuters.
Biden called the Sunday Kyiv strikes acts of "barbarism", as leaders from the Group of Seven nations met for a summit in Germany.