Ukraine battled Russian forces in its eastern region Tuesday while overnight missile strikes hit the southern port of Odessa, as the United States warned that Russian President Vladimir Putin is ready for a long war.

Washington's bleak prediction came after Ukraine said its membership of the European Union was a question of "war and peace" for the whole continent as it faces up to Russia, well over two months after Moscow invaded.

Kyiv also hailed what it said was EU powerhouse Germany's change of stance on a Russian oil embargo and on supplying arms to Ukraine.

Violence raged in the south, where the missile strikes in Odessa overnight destroyed buildings, set ablaze a shopping centre and killed one person, just hours after a visit by European Council President Charles Michel.

Officials also said some 1,000 troops remain trapped in increasingly dire circumstances in the Azovstal steelworks at the devastated city of Mariupol.

Russia invaded Ukraine on February 24 -- but Ukrainian forces managed to push Moscow's forces back from Kyiv. The capital's mayor said Tuesday that two-thirds of its residents have returned.

Putin has given few hints on his plans, but US Director of National Intelligence Avril Haines said Tuesday that the Russian leader will not end the war with the Donbas campaign and is determined to build a land bridge to the Russian-controlled territory in Moldova.

US intelligence also views it as increasingly likely that Putin will mobilise his entire country, including ordering martial law, and is counting on his perseverance to wear down Western support for Ukraine.

Moscow switched its focus to the Russian-speaking Donbas region in the east, where separatists have been fighting since 2014, after failing to take Kyiv.

Ukraine's presidency said the "epicentre of the fighting has moved" to Bilogorivka in the Lugansk region of the Donbas, the site of a deadly Russian air strike Sunday that Ukrainian officials said killed 60 people.

Shelling also continued in Ukraine's easternmost strongholds, the sister cities of Severodonetsk and Lysychansk, it said.

Ukraine's President Volodymyr Zelensky said Russian forces were being "pushed from" Kharkiv -- but that was tempered by a revelation by the region's governor that 44 civilian bodies had been found under the rubble of a destroyed building in the eastern town of Izyum, now under Russian control.

His counterpart in Donetsk said three civilians were killed in the region on Tuesday.

Civilians were struggling to survive between the constantly shifting front lines.

"I feel total apathy. I am morally starved -- not to mention physically," said bricklayer Artyom Cherukha, 41, as he collected water trickling from a natural spring in Lysychansk.

He was trying to get supplies for his family of nine, as people in the area steadily lose access to water and food.

"We sit here counting the bombs," said Cherukha.

Russia's defence ministry said it hit 74 targets on Tuesday and downed a Ukrainian drone above the strategic Snake Island in the Black Sea.

Ukraine has been pushing Western countries for more support, and has been particularly critical of Germany for its slow response and unwillingness to give up Russian energy.

The tone changed on Tuesday with German Foreign Minister Annalena Baerbock's surprise visit to Bucha, a town outside Kyiv where Russian troops have been accused of war crimes.

"I would like to thank Germany for changing its position on a number of issues" including arms supplies to Kyiv and supporting a Russian oil embargo, Ukrainian Foreign Minister Dmytro Kuleba told reporters in Kyiv with Baerbock.

Kuleba pushed for the European Union to admit his country.

"Ukraine's membership in the EU is a matter of war and peace in Europe," said Kuleba. "One of the reasons that this war started is that Putin was convinced that Europe doesn't need Ukraine."

US President Joe Biden has meanwhile resurrected a World War II measure to aid Kyiv, opening the spigots on artillery, anti-aircraft missiles, anti-tank weapons and other powerful materiel.

On Tuesday US lawmakers were to debate a nearly $40 billion aid package, which is expected to pass comfortably with rare bipartisan support.

Western powers on Tuesday separately accused Russian authorities of carrying out a cyberattack against a satellite network an hour before the invasion of Ukraine to pave the way for its assault.

Moscow has made more progress in southern Ukraine but more than a thousand Ukrainian soldiers remain in Mariupol's Azovstal steelworks, Deputy Prime Minister Iryna Vereshchuk told AFP.

The plant is the final bastion of Ukrainian resistance in the city, which has seen relentless destruction.

An online petition calling on the United Nations to extract all remaining soldiers garnered more than 1.1 million signatures Tuesday.

"Hundreds are injured. There are people with serious injuries who require urgent evacuation," said Vereshchuk.

Many civilians have been evacuated from the plant in recent days, as Russia pushes for full control of Mariupol to open up a land corridor from Crimea, which it seized in 2014.

Zelensky said that Kravchuk, who lived through Nazi occupation during World War II as a boy, understood the horror of war.

"He wished for peace for Ukraine with all his heart," Zelensky said.

"I am sure that we will make that happen, we will achieve our victory and our peace."