The "powerful" air strikes in the west came hours after President Volodymyr Zelensky accused Moscow of wanting to "destroy" the entire eastern region of Donbas near the border with Russia.

During its latest briefing, Russia's defence ministry said it had hit over a dozen military targets with air strikes at various locations across Ukraine.

Following the attack on Lviv, black smoke billowed from the gutted roof of a car repair shop above the railway tracks in the northwest of the city as air raid sirens wailed.

"Fires were set off as a result of the strikes. They are still being put out. The facilities were severely damaged," the Lviv regional governor Maksym Kozytsky said on social media.

In the south, Russia continued its push to capture the besieged city of Mariupol where the last remaining Ukranian forces in the strategic port prepared for a final stand.

Ukraine has pledged to fight on and defend the strategic port city, defying a Russian ultimatum issued Sunday that called on the remaining fighters inside the encircled Azovstal steel plant to lay down their arms and surrender.

Mariupol has become a symbol of Ukraine's unexpectedly fierce resistance since Russian troops invaded the former Soviet state on February 24.

"The city still has not fallen," Prime Minister Denys Shmyhal said.

"There's still our military forces, our soldiers. So they will fight to the end," he told ABC's "This Week".

"We will not surrender."

While several large cities were under siege, he said, not one -- with the exception of Kherson in the south -- had fallen, and more than 900 towns and cities had been re-captured.

Capturing Mariupol would allow Russia to have a land bridge between the Crimea peninsula, which it annexed in 2014, and the two Moscow-backed separatist statelets in Ukraine's east.

- 'Last chance to save you' -

In the east, Ukrainian authorities urged people in Donbas to move west to escape a large-scale Russian offensive to capture its composite regions of Donetsk and Lugansk.

"Russian troops are preparing for an offensive operation in the east of our country in the near future. They want to literally finish off and destroy Donbas," Zelensky said in a statement late Sunday.

Lugansk governor Sergiy Gaiday said the coming week would be "difficult".

"It may be the last time we have a chance to save you," he wrote on Facebook.

Heavy bouts of shelling also resumed in the country's second city of Kharkiv on Monday morning, according to an AFP reporter in the city.

The shelling comes a day after at least five people were killed and 20 wounded in during a string of strikes in the city just 21 kilometres (13 miles) from the Russian border on Sunday.

Maksym Khaustov, the head of the Kharkiv region's health department, confirmed the deaths there following the strikes that had ignited fires throughout the city and torn roofs from buildings.

"The whole home rumbled and trembled," 71-year-old Svitlana Pelelygina told AFP as she surveyed her wrecked apartment. "Everything here began to burn."

- 'Catastrophe' -

Ukraine officials also said on Monday they were halting the evacuation of civilians from frontline town and cities in the east of the country for the second consecutive day, accusing Russian forces of blocking and shelling escape routes.

"Unfortunately, today, April 18, there will be no humanitarian corridors. In violation of international humanitarian law, the Russian occupiers have not stopped blocking and shelling humanitarian routes," Deputy Prime Minister Iryna Vereshchuk said in a statement on social media.

But Lugansk governor Gaiday announced earlier that he had proceeded with evacuations.

"At our own peril and risk, we took out several dozen people anyway, but it's already dangerous," he told Ukrainian media.

During an interview with CNN broadcast on Sunday, Zelensky said he had invited his French counterpart Emmanuel Macron to visit Ukraine to see for himself evidence that Russian forces have committed "genocide" -- a term Macron has avoided.

"I talked to him yesterday," Zelensky told CNN in an interview recorded on Friday and broadcast Sunday.

"I just told him I want him to understand that this is not war, but nothing other than genocide. I invited him to come when he will have the opportunity. He'll come and see, and I'm sure he will understand."

Zelensky, describing the situation in Mariupol as "inhuman", has called on the West to immediately provide heavy weapons -- a request he frequently airs.

But Russia has warned the United States this week of "unpredictable consequences" if it sent its "most sensitive" weapons systems to Ukraine.

Its defence ministry claimed Saturday to have shot down a Ukrainian transport plane in the Odessa region that was carrying weapons supplied by Western nations.

On Sunday, spokesman Igor Konashenkov said Russian missiles had destroyed ammunition, fuel and lubricant depots in eastern Ukraine and 44 Ukrainian military facilities, including command posts.

Russian air defence systems shot down two Ukrainian MiG-29 aircraft in the Kharkiv region and a drone near the city of Pavlograd, he added.

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