KYIV: The Ukrainian president said he would not surrender "a single centimetre" in battles over Donetsk, while Russian-installed officials claimed Ukrainian tanks were moving into a southern city.

The conflict in the industrial area of ​​Donetsk centres around the towns of Bakhmut, Soledar and Avdivka. The region has seen some of the heaviest fightings since Russian troops invaded Ukraine in late February.

In a video message on Tuesday night, Zelenskiy said that the activities of the occupiers are at a very high level and they are carrying out dozens of attacks every day.

A Russian-installed mayor in the town of Snihurivka, east of the southern city of Mykolaiv, was cited by Russia's RIA news agency as saying residents had seen tanks and that fierce fighting was going on.

Kirill Stremousov, deputy head of the Russian-installed administration in the Kherson region, said on the Telegram messaging service that Ukrainian forces had tried to advance on three fronts, including Snihurivka.

Vitaly Kim, the Ukrainian governor of Mykolaiv region, apparently quoting an intercepted conversation between Russian servicemen, suggested that Ukrainian forces had already pushed the Russians out of the area.

There was no official word on the situation in the town from military officials in either Ukraine or Russia.

Ukrainian forces have been on the offensive in recent months while Russia is regrouping to defend areas of Ukraine it still occupies, having called up hundreds of thousands of reservists over the past month.

Kyiv-based military analyst Oleh Zhdanov said on Tuesday that 21 Russian conscripts had surrendered to Ukrainian forces around Svatove in Luhansk region.

Russia's Tass news agency reported that Ukrainian forces fired Himars multiple launch rockets at Stakhanov, a city in Luhansk.

According to the report, four civilians were killed in the rocket attack on Sunday night.

Giving an update on the situation in the southern region of Kherson, the Ukrainian military on Tuesday evening accused Russian troops of more looting and destroying infrastructure. A showdown has been looming for weeks in Kherson city, the only regional capital Russia has captured intact since its invasion.

"A convoy of trucks passed over the dam of the Kakhova hydroelectric station loaded with home appliances and building materials," the military said.

In Kherson city, it said Russian troops removed exhibits, furniture and equipment from a museum devoted to the painter Oleksiy Shovkunenko.

Zelenskiy also said in his address that about 4 million people were without power in 14 regions plus the capital Kyiv. Scheduled hourly power outages would affect the whole of the country on Wednesday, Ukraine's electrical grid operator Ukrenergo said.

Russian forces targeted Ukraine's energy infrastructure with missiles and drones in recent weeks as winter approaches when mean temperatures typically drop to several degrees below zero Celsius (32 Fahrenheit), with lows of minus 20C.

The U.N. The General Assembly is due to vote next week on a draft resolution recognizing that Russia must be responsible for reparation in Ukraine for the injury, including any damage, caused by "internationally wrongful acts".

Three-quarters of the 193-member General Assembly denounced Russia's invasion in a vote in March, and in October condemned its self-proclaimed annexation of parts of Ukraine.