(Adds report of shelling from Kyiv mayor; U.S. defense secretary on hypersonic missiles)
By Pavel Polityuk
LVIV, Ukraine (Reuters) - Russian and Ukrainian forces fought for the Ukrainian port city of Mariupol on Sunday, where residents are trapped with little food, water and power, while Ukraine's president appealed to Israel for help in pushing back Russia's assault.
In the capital, Kyiv, shellfire hit several homes and a shopping centre in the Podil district late on Sunday, killing at least one person, the city's mayor said.
In his latest appeal for help from abroad, Ukrainian President Volodymyr Zelenskiy addressed the Israeli parliament by video link and questioned Israel's reluctance to sell its Iron Dome missile defence system to Ukraine.
"Everybody knows that your missile defence systems are the best... and that you can definitely help our people, save the lives of Ukrainians, of Ukrainian Jews," said Zelenskiy, who is of Jewish heritage.
Israeli Prime Minister Naftali Bennett has held numerous calls with both Zelenskiy and Russian President Vladimir Putin to try to end the conflict.
Mariupol has suffered some of the heaviest bombardments since Russia invaded Ukraine on Feb. 24. Many of its 400,000 residents remain trapped with little if any food, water and power.
Burying his neighbours in a makeshift grave by the roadside, a man who identified himself as Andrei said they had died not by shelling but of ailments, stress and cold after weeks without access to medical help.
Fighting continued inside the city on Sunday, regional governor Pavlo Kyrylenko said, without elaborating.
Russia called on Ukrainian forces in Mariupol to lay down their arms, saying a "terrible humanitarian catastrophe" was unfolding.
It said defenders who did so were guaranteed safe passage out of the city and humanitarian corridors would be opened from 1000 Moscow time (0700 GMT) on Monday.
Ukrainian Deputy Prime Minister Iryna Vereshchuk said over 7,000 people were evacuated from Ukrainian cities through humanitarian corridors on Sunday, more than half of them from Mariupol. She said the government planned to send nearly 50 buses to Mariupol on Monday for further evacuations.
Russia and Ukraine have made agreements throughout the war on humanitarian corridors to evacuate civilians, but have accused each other of frequent violations of those.
Mariupol city council said on Telegram on Saturday that several thousand residents had been "deported" to Russia over the past week. Russian news agencies said buses had carried hundreds of refugees from Mariupol to Russia in recent days.
U.S. ambassador to the United Nations Linda Thomas-Greenfield told CNN the deportation accounts were "disturbing" and "unconscionable" if true, but said Washington had not yet confirmed them.
Russian forces bombed an art school on Saturday in which 400 residents were sheltering, but the number of casualties was not yet known, Mariupol's council said.
Reuters could not independently verify the claims. Russia denies targeting civilians.
On Saturday, Zelenskiy called the siege of Mariupol a war crime and "a terror that will be remembered for centuries to come."
Greece's consul general in Mariupol, the last EU diplomat to evacuate the city, said it was joining the ranks of places known for having been destroyed in wars.
"What I saw, I hope no one will ever see," he said.
FEW ADVANCES
Capturing Mariupol would help Russian forces secure a land corridor to the Crimea peninsula that Moscow annexed from Ukraine in 2014.
Putin says Russia's "special operation" is aimed at disarming Ukraine and rooting out dangerous nationalists. Western nations call it an aggressive war of choice and have imposed punishing sanctions aimed at crippling Russia's economy.
Ukraine and its Western backers say Russian ground forces have made few advances in the last week, concentrating instead on artillery and missile strikes.
Zelenskiy's adviser Oleksiy Arestovych said on Sunday there had been a relative lull over the past 24 hours, with "practically no rocket strikes on cities". He said front lines were "practically frozen".
Later on Sunday, however, Kyiv mayor Vitali Klitschko reported several explosions in the capital's Podil district and said rescue teams were putting out a large fire at the shopping centre. He said at least one person was killed.
Reuters was not able to immediately verify the report.
The U.N. human rights office said at least 902 civilians had been killed as of midnight Saturday, though the real toll was probably much higher. Ukrainian prosecutors said 112 children had been killed.
The U.N. refugee agency said 10 million Ukrainians have been displaced, including some 3.4 million who have fled to neighbouring countries such as Poland. Officials in the region said they were reaching capacity to comfortably house refugees.
Russia said it had launched cruise missiles from ships in the Black Sea and Caspian Sea, as well as hypersonic missiles from Crimean airspace. Hypersonic missiles travel faster than five times the speed of sound and their speed, manoeuvrability and altitude make them difficult to intercept.
Russia deployed them for the first time in Ukraine on Saturday in a strike it said destroyed an underground depot for missiles and aircraft ammunition.
U.S. Defense Secretary Lloyd Austin told CBS News' "Face the Nation" program on Sunday Putin could have resorted to such weapons "because he's trying to reestablish some momentum", but said he didn't see them as a game-changer.
Foreign Minister Mevlut Cavusoglu of Turkey, which like Israel has tried to mediate, said the sides were getting closer to agreement on "critical" issues.
Kyiv and Moscow reported some progress last week toward a political formula that would guarantee Ukraine's security, while keeping it outside NATO - a key Russian demand - though each side accused the other of dragging things out.
In the southern city of Kherson, video seen by Reuters showed dozens of protesters, some wrapped in Ukraine's blue-and-yellow flag, chanting "Go home" in Russian at two military vehicles with Russian markings. The vehicles turned and left.
"I want the war to be over, I want them (Russian forces) to leave Ukraine in peace," said Margarita Morozova, 87, who survived Nazi Germany's siege of Leningrad in World War Two and has lived in Kharkiv, eastern Ukraine, for the past 60 years.