Ukrainians in the city of Pokrovsk, Donbas region, are fearing the worst as Russia looks set to make significant advances there.
Kyiv struck a blow to Moscow with its recent into burst the Russian Kursk region.
But there are fears among some Ukrainians that this may not be enough to stop Russian soldiers from advancing in parts of eastern Ukraine.
In Pokrovsk, a police officer told Al Jazeera: “We’re expecting a nightmare.”
The comments come after Ukraine ordered families with children in the city to evacuate on Monday.
Donetsk region head Vadym Filashkin said over 53,000 people, including almost 4,000 children, are in the city.
He added: “When our cities are within range of virtually any enemy weapon, the decision to evacuate is necessary and inevitable.”
Russia has also claimed to have taken control over the town of Niu-York in Donetsk.
The town is important as it is a Ukrainian logistics hub. The Russian Defence Ministry said it had “liberated” the town.
This comes after Ukrainian President Volodymyr Zelensky said Ukraine had gained control of over 1,250 square kilometres and 92 settlements in Kursk.
He said: “The Russian border area opposite our Sumy region has been mostly cleared of Russian military presence.
“A few months ago, many people around the world would have said this was impossible and crossed Russia’s strictest ‘red line’.”
While the situation for Ukraine is difficult on home turf, it is continuing to hit Russians in Kursk.
Ukraine has captured hundreds of Russian men near the region, according to reports.