A local resident walks next to a tank of pro-Russian troops in Mariupol, Ukraine, on Tuesday. Reuters

Russia hurled its military might against Ukrainian cities and towns and poured more troops into the war, seeking to slice the country in two in a potentially pivotal battle for control of the eastern industrial heartland of coal mines and factories.

The fighting on Tuesday unfolded along a boomerang-shaped front hundreds of miles long in what is known as the Donbas. If successful, it would give President Vladimir Putin a victory following the failed attempt by Moscow's forces to storm the capital, Kyiv, and heavier-than-expected casualties.

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Meanwhile, Russia gave Ukrainian fighters still holding out in Mariupol a fresh ultimatum to surrender on Wednesday as it pushed for a decisive victory in its new eastern offensive, while Western governments pledged more military help to Kyiv.

A crater and a destroyed home are pictured in the village of Yatskivka, eastern Ukraine. File/AFP

Thousands of Russian troops backed by artillery and rocket barrages were advancing in what Ukrainian officials have called the Battle of the Donbas.

In Mariupol, the now-devastated port city in the Donbas, Ukrainian troops said the Russian military dropped heavy bombs to flatten what was left of a sprawling steel plant and hit a hospital where hundreds were staying.

The eastern cities of Kharkiv and Kramatorsk came under deadly attack. Russia also said it struck areas around Zaporizhzhia and Dnipro west of the Donbas with missiles.

Rescuers work at a residential building damaged during Ukraine-Russia conflict in Mariupol, Ukraine. Reuters

Russian Defense Ministry spokesman Maj. Gen. Igor Konashenkov said Moscow's forces bombarded numerous Ukrainian military sites, including troop concentrations and missile-warhead storage depots, in or near several cities or villages. Those claims could not be independently verified.

Russia's nearly eight-week-long invasion has failed to capture any of Ukraine's largest cities, forcing Moscow to refocus in and around separatist regions.

The biggest attack on a European state since 1945 has, however, seen nearly 5 million people flee abroad and reduced cities to rubble.

Russia was hitting the Azovstal steel plant, the main remaining stronghold in Mariupol, with bunker-buster bombs, a Ukrainian presidential adviser said late on Tuesday. Reuters could not verify the details.

A view shows damaged buildings during Ukraine-Russia conflict in Mariupol, Ukraine, on Tuesday. Reuters

Both sides have described the assault that began Monday as a new phase of the war.

Ukrainian President Volodymyr Zelensky said the Russian military was throwing everything it has into the battle, with most of its combat-ready forces now concentrated in Ukraine and just across the border in Russia.

"They have driven almost everyone and everything that is capable of fighting us against Ukraine,” he said in his nightly video address to the nation.

Despite claims that they are hitting only military sites, the Russians continue to target residential areas and kill civilians, he said.

Local residents push a cart with a child past buildings destroyed during Ukraine-Russia conflict in Mariupol. Reuters

"The Russian army in this war is writing itself into world history forever as the most barbaric and inhuman army in the world,” Zelensky said.

Weeks ago, after the abortive Russian push to take Kyiv, the Kremlin declared that its main goal was the capture of the mostly Russian-speaking Donbas, where Moscow-backed separatists have been fighting Ukrainian forces for eight years.

A Russian victory in the Donbas would deprive Ukraine of the industrial assets concentrated there, including mines, metals plants and heavy-equipment factories.