Moscow said on Sunday that its military had struck a supply of weapons sent to Ukraine by Western nations near Odessa.
The attack came as U.S. House of Representatives Speaker Nancy Pelosi met Ukrainian President Volodymyr Zelensky in Kyiv and vowed to continued support for that country's fight against Russia.
Footage posted by Zelensky on Twitter on Sunday showed him, flanked by an armed escort and dressed in military fatigues, greeting a congressional delegation led by Pelosi outside his presidential office the previous day.
"Our delegation travelled to Kyiv to send an unmistakable and resounding message to the entire world: America stands firmly with Ukraine," Pelosi, the highest-ranking U.S. official to visit Ukraine since Russia invaded on Feb. 24, said in a statement. She condemned Russian President Vladimir Putin's "diabolic invasion".
Moscow calls its actions a "special military operation" to disarm Ukraine and rid it of anti-Russian nationalism fomented by the West. Ukraine and the West say Russia launched an unprovoked war of aggression.
Western nations have imposed broad economic sanctions on Russia and have been shipping increasing quantities of weapons to help Ukraine defend itself.
Pelosi said on Friday she hoped to pass a $33 billion aid package for Ukraine that President Joe Biden has requested "as soon as possible".
Russia's defense ministry said on Sunday it had carried out a missile strike on a military airfield near the port city of Odesa, destroying a runway and a hangar containing weapons and ammunition supplied to Ukraine by the United States and European countries.
On Saturday, Ukraine said Russian missiles had knocked out a newly constructed runway at Odesa's main airport.
It was unclear if they were referring to the same incident and Reuters could not immediately verify the reports.
Some civilians leave the Mariupol plant
Moscow has turned its focus to Ukraine's south and east after failing to capture Kyiv in the early weeks of a war that has flattened cities, killed thousands of civilians and forced more than 5 million to flee the country.
Russia's defense ministry accused Ukraine's forces of shelling a school, kindergarten and cemetery in villages in the occupied southern Kherson region, the Russian RIA news agency said on Sunday.
The ministry said civilians had been killed and wounded but gave no further details. There was no immediate response from Ukraine and Reuters could not independently verify the report.
Russian forces captured the town of Kherson, 100 km (60 miles) north of Russian-annexed Crimea, in March and since then have mostly occupied Mariupol, a strategic eastern port city on the Azov Sea.
Ukraine's military said in a bulletin on Sunday that Russian forces were fighting to break beyond Kherson's administrative borders and prepare the way for attacks on the cities of Mykolayiv and Kryvyi Rih.
In Mariupol, Russia declared victory on April 21 even as hundreds of holdout Ukrainian troops and civilians took shelter in the city's bombed-out Azovstal steelworks, where they have been trapped with little food, water or medicine.
On Saturday, a Ukrainian fighter inside said that 20 women and children had made it out of the sprawling plant.
"We are getting civilians out of the rubble with ropes - it's the elderly, women and children," said the fighter, Sviatoslav Palamar.
Palamar said Russia and Ukraine were respecting a local ceasefire. Meanwhile, Russia's defense ministry said on Sunday that 46 civilians had left the area around the steel plant the previous day.
Speaking to thousands of people in St. Peter's Square for his noon blessing on Sunday, Pope Francis said Mariupol had been "barbarously bombarded and destroyed" in a war he called a "macabre regression of humanity".
In the east, Moscow is pushing for complete control of the Donbas region, where Russian-backed separatists already controlled parts of Luhansk and Donetsk provinces before the invasion. Russian forces carried out missile strikes across the south and east on Saturday, Ukrainian officials said.
On Sunday, Kharkiv region governor Oleh Synehubov warned residents in the north and east of the city of Kharkiv to remain in their shelters due to heavy Russian shelling. Reuters could not immediately verify reports of shelling in the area.
Serhiy Gaidai, the governor of the Luhansk region, in a post on social media on Sunday, urged people to evacuate while it was still possible.
Zelensky said in a late-night video address on Saturday that Russia was "gathering additional forces for new attacks against our military in the east of the country" and "trying to increase pressure in the Donbas".
Despite weeks of peace talks, both sides looked to be as far apart as ever on Saturday.
Ukraine accuses Russian troops of carrying out atrocities in areas near Kyiv from which they have since been pushed back in early April, a claim denied by Moscow. Negotiators last met face-to-face on March 29 and have since spoken by video link.