Ukraine has reported intense Russian shelling across the front lines as both sides traded blame for the weekend strike on the Zaporizhzhia nuclear complex which triggered international concern about a potential atomic disaster.
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Heavy fighting was reported on Tuesday in frontline towns near the eastern city of Donetsk, where Ukrainian officials said Russian troops were launching waves of attacks as they try to seize control of the industrialised Donbas region.
"The situation in the region is tense - shelling is constant throughout the front line ... The enemy is also using air strikes a great deal," Donetsk regional Governor Pavlo Kyrylenko told Ukrainian television.
"The enemy is having no success. Donetsk region is holding."
Around Kharkiv in the northeast, Ukrainian troops captured the town of Dovhenke from Russian occupiers and were advancing towards Izium, Ukrainian Presidential adviser Oleksiy Arestovych said in a video posted on YouTube.
The daily battlefield report from Ukraine's military general staff said towns north, east and southeast of Kharkiv had come under fire from tanks, artillery and rockets.
"The situation is very interesting. Ukraine's forces are moving very successfully. Attempts by Russia to regain lost ground were not successful. Ukraine may end up encircling them," Arestovych said.
In the southeast, the key Antonovskyi bridge over the Dnipro river in Kherson region was targeted again by Ukrainian forces trying to disrupt Russian supply lines.
Yuri Sobolevsky, deputy head of Kherson regional council ousted by Russian occupation forces, said on Telegram the bridge had been seriously damaged after "overnight actions".
Reuters was unable to verify the reports.
US Under Secretary of Defence for Policy Colin Kahl on Monday said Russia had suffered between 70,000 and 80,000 casualties, either killed or wounded, since its invasion of Ukraine began on February 24. Russia calls the war a "special military operation".
United Nations chief Antonio Guterres on Monday called any attack on a nuclear plant "suicidal" and demanded UN nuclear inspectors be given access to Zaporizhzhia, the largest nuclear power complex of its kind in Europe.
Russia's invading forces seized the southern Ukrainian region containing Zaporizhzhia in March, when the site was struck without damage to its reactors. The area, including the city of Kherson, is now the target of a Ukrainian counter-offensive.
Ukraine appealed for the area around the complex to be demilitarised and for the International Atomic Energy Agency, the UN's nuclear watchdog, to be let in. Russia said it too favoured an IAEA visit, which it accused Ukraine of blocking.
Both sides blamed the other for weekend attacks around the complex, which is still being run by Ukrainian technicians. Ukraine said three radiation sensors were damaged and two workers injured by shrapnel.
In an evening video shared online, Ukrainian President Volodymyr Zelenskiy called for new Western sanctions on Russia's nuclear industry "for creating the threat of a nuclear disaster."
Stepping up its fiscal aid and military spending on Ukraine, Washington announced it will send $US4.5 billion ($A6.4 billion) in budgetary support and $US1 billion ($A1.4 billion) in weapons, including long-range rocket munitions and armoured medical transport vehicles.
Overall, the United States has contributed more than $US18 billion ($A26 billion) to Ukraine this year.
While pouring arms and money into Ukraine, the US was also enforcing financial sanctions against the Kremlin and the wealthy elites who support President Vladimir Putin.
Russia says it is waging a "special military operation" in Ukraine to rid it of nationalists and protect Russian-speaking communities. Ukraine and the West describe Russia's actions as an unprovoked war of aggression.
The conflict has displaced millions of people, killed thousands and left cities, towns and villages in ruins.
Australian Associated Press
