Two people were killed, according to Ukrainian officials, on the second day of Russian attacks on central Kherson, a city that Ukraine recaptured last month.
The regional governor said the heavy bombardment of critical infrastructure in the port area left the entire city without electricity.
The withdrawal of Russian forces from Kherson was one of the biggest setbacks for Moscow since the February invasion.
For weeks, energy facilities in Ukrainian cities have been targeted.
Millions of Ukrainians are left without heat or power, with winter temperatures dropping below freezing.
Officials said the shells landed 100 meters from the main administration building in the city of Kherson, a day after the building itself was badly damaged. Ukrainian media said a 32-year-old paramedic and a 70-year-old man were killed in the attack, which hit a medical aid point.
Explosions also occurred in Kharkiv, Ukraine's second largest city. Mayor Igor Terekhov said Russia was bombing infrastructure facilities and appealed to residents to stay in shelters if possible.
Volker Türk, the UN High Commissioner for Human Rights, warned that the Russian attacks were subjecting millions of Ukrainians to "extreme hardship" and that further attacks on energy facilities could "lead to a further serious deterioration of the humanitarian situation and lead to further displacement".
Meanwhile, Russian-backed officials in eastern Ukraine said Ukrainian forces had launched their "largest strike" on Russian-controlled central Donetsk since 2014, when separatists sparked a conflict by seizing parts of the Donbass region.
The official appointed from Russia, Alexei Kolmzin, said 40 rockets were fired, killing one person and wounding nine others.
It was not possible to confirm the details of the attack, but Colmzin posted pictures of destroyed buildings in the city.
In his speech to the United Nations Human Rights Council in Geneva, Volker Türk said the war had left 18 million people in need of humanitarian aid.
It detailed summary executions of civilians by the Russian military between February and April, including horrific killings in Bucha, a town outside kyiv
On Thursday, Ukrainian President Volodymyr Zelensky appealed to European Union leaders to help defeat "Russian energy terrorism" by preserving Ukraine's energy supply with about 2 billion cubic meters of gas and electricity worth about 800 million euros worth of electricity.
He said that Ukraine has achieved tangible victories and has begun to build an air shield for Ukraine. The president said that the capital Kyiv was also targeted by 13 drones on Wednesday, but the army managed to repel them.