Britain said Russian forces had widened strikes on civilian infrastructure following battlefield setbacks and were likely to expand their targets further.
The Ukrainian military said on Sunday that its forces repelled attacks by Russian troops in the areas of the Kharkiv region in the east and Kherson in south where Ukraine launched counter-offensives this month, as well as in parts of Donetsk in the southeast.
It said Ukrainian troops had advanced to the eastern bank of the Oskil River in Kharkiv region.
"From yesterday, Ukraine controls the east bank," it said on Telegram. Serhiy Gaidai, governor of the neighbouring Luhansk region, said this meant the "deoccupation" of his region was "not far away."
In his nightly address on Sunday, Zelenskiy vowed to keep up the pressure on Moscow.
"Perhaps it seems to some of you that after a series of victories we now have a lull of sorts," he said in his regular nightly address. "But there will be no lull. There is preparation for the next series ... For Ukraine must be free. All of it."
Ukrainians who returned to the northeastern area retaken in Kyiv's lightning advance earlier this month were searching for their dead while Russian artillery and air strikes kept pounding targets across Ukraine's east.
On Friday, Russian President Vladimir Putin brushed off Ukraine's swift counteroffensive and said Moscow would respond more forcefully if its troops were put under further pressure.
Such repeated threats have raised concerns Putin could at some point turn to small nuclear weapons or chemical warfare.
US President Joe Biden, asked what he would tell Putin if he was considering using such weapons, replied in an interview with CBS's 60 Minutes: "Don't. Don't. Don't. It would change the face of war unlike anything since World War Two."
Some military analysts have said Russian might also stage a nuclear incident at Zaporizhzhia, Europe's largest nuclear power plant held by Russia but run by Ukrainian staff.
Moscow and Kyiv have accused each other for shelling around the plant that has damaged buildings and disrupted power lines needed to keep it cooled and safe.
US Army General Mark Milley, chairman of the Joint Chiefs of Staff, called for vigilance on Sunday after visiting a base in Poland aiding Ukraine's war effort.
"The war is not going too well for Russia right now So it's incumbent upon all of us to maintain high states of readiness, alert," he said.
In a intelligence update, Britain's defence ministry said Russian strikes at civilian infrastructure, including a power grid and a dam, had intensified.
"As it faces setbacks on the front lines, Russia has likely extended the locations it is prepared to strike in an attempt to directly undermine the morale of the Ukrainian people and government," it said.
On Saturday, Zelenskiy said authorities had found a mass grave containing the bodies of 17 soldiers in Izium, some of which he said bore signs of torture.
Ukrainian officials said last week they had found 440 bodies in woods near Izium. They said most of the dead were civilians and the causes of death had not been established.
The Kremlin has not commented on the discovery of the graves.
