Russian missiles struck near Lviv's airport in the far west of Ukraine early Friday, as Moscow expanded a countrywide aerial bombardment campaign that has intensified allegations of war crimes and deliberate targeting of civilians.

Lviv Mayor Andriy Sadovyi said Russian forces had destroyed an aircraft repair plant — which sits near the Polish border.

Ambulance and police vehicles raced to the scene, while motorists were turned away at checkpoints. An AFP reporter saw a thick pall of smoke billowing over the airport.

“Several missiles hit an aircraft repair plant,” the mayor said on the messaging app Telegram, adding that the plant had been destroyed.

Located 70 kilometres (45 miles) from the border, Lviv had until now largely escaped military strikes from Russian forces.

But as Russian President Vladimir Putin's three-week-old ground offensive has stalled under fierce Ukrainian resistance, Moscow has increasingly turned to air and long-range strikes to gain the upper hand.

According to Pentagon estimates, Russia has now fired over 1,000 missiles at Ukrainian targets since the war began.

Before dawn broke on Friday, air raid alarms had rung in cities across the country.

Ukraine's government listed a kindergarten and market in the eastern city of Kharkiv among the latest targets.

Several cities from Sumy in the north to Mariupol in the south are virtually under siege — cut off from supplies and facing near-constant attack.

In his latest night-time video message, Ukrainian President Volodymyr Zelensky admitted the situation in several Ukrainian cities was “difficult.” But, he said, “we will not leave you behind and we will not forgive them. You will be free.”

Rescue effort

In the besieged southern city of Mariupol, rescue workers were still searching desperately for any survivors buried beneath the rubble of a bombed-out theatre, amid fears that hundreds may be trapped.

Twenty-four hours after the once-gleaming whitewashed theatre was hollowed out by a Russian strike, the number of dead, injured or trapped is still unclear.

Ukraine's ombudswoman Lyudmyla Denisova said a bomb shelter in the building had survived the impact, and some “adults and children” had emerged alive.

“Work is under way to unlock the basement,” she said.

The attack on a civilian building marked with the words “DETI”, or “children” in Russian, has sparked a wave of international revulsion and heaped pressure on Russia's few remaining allies — most notably China — to condemn Moscow's apparent deliberate targeting of civilians.

In a call later on Friday, US President Joe Biden is set to warn his counterpart Xi Jinping that Beijing will face “costs” for “any actions it takes to support Russia's aggression”, according to US Secretary of State Antony Blinken.

Biden hoped China would use “whatever leverage they have to compel Moscow to end this war”, the top US diplomat said.

In the wake of the theatre attack, Blinken also said it was “difficult to conclude” that Putin's regime had not engaged in war crimes by targeting civilians.

'It is hell'

Russia has routinely denied such allegations and the ministry of defence has claimed it did not strike any ground targets in Mariupol on the afternoon the theatre was hit.

It instead suggested Ukraine's hardline nationalist Azov battalion, a frequent target of Russian propaganda, mined the theatre and held civilian hostages there in “a new bloody provocation”.

Russia's siege of the city — cutting power, as well as many communications links and food supplies — has closed access and made independent verification all but impossible.

Local officials say more than 2,000 people have died so far in indiscriminate shelling, and 80 percent of its housing has been destroyed.

“In the streets, there are the bodies of many dead civilians,” Tamara Kavunenko, 58, told AFP after fleeing the city.

“It's not Mariupol anymore,” she said. “It is hell.” Ukrainian MP Sergiy Taruta said that Russian forces' blockade of the city, killing of medics and destroying emergency equipment was stymying rescue efforts.

He claimed some people had emerged from the wreckage, but warned: “All those who survived the bombing will either die under the rubble of the theatre or have already died.”

Italy's Minister of Culture Dario Franceschini said his country was ready to rebuild the theatre “as soon as possible”.

Battle for the capital

Outgunned and outnumbered, Zelensky has beseeched allies to provide more military assistance — even as an arsenal of anti-tank and anti-aircraft missiles flood into the country.

Slovakia confirmed it is willing to provide powerful Russian-made S-300 anti-aircraft missile system to Ukraine, but only on the condition that it receive a substitute from Nato allies.

On Wednesday, Zelensky told German lawmakers that Russia was throwing up another “Berlin Wall”, a dividing line between “freedom and bondage” in Europe.

“And this wall is growing bigger with every bomb,” he added.

That dividing line is currently drawn around 15 kilometres from Kyiv, where Russian troops are still trying to surround the capital in a slow-moving offensive.

What remains of the city's 3.5 million population has been bracing for a full-on Russian assault for weeks and is now living with the terror of daily artillery attacks.

Iryna Voinovska, 55, recently awoke at five in the morning to a whistle and the sound of her husband's screams.

Debris from a downed missile had blown a huge chunk out of the top of her apartment block.

“The main thing is that we are alive ... Unfortunately, one woman died on the 16th floor, she was crushed by a concrete slab,” she told AFP.

In Odessa, on the Black Sea, civilians were bracing for attack, with tanks deployed at intersections and monuments covered in sandbags.

“Our beautiful Odessa,” said Lyudmila, an elegant elderly woman wearing bright lipstick, as she looked apologetically at her city's empty, barricaded streets.

“But thank God we are holding on! Everyone is holding on! “