With the war poised to enter its third month on Sunday, Ukrainian authorities said "fierce battles" were raging in the east and the United Nations said nearly 5.2 million people had fled the country.
Ukrainian President Volodymyr Zelensky called for a meeting with Russian counterpart Vladmir Putin "to end the war", which began with a full-scale Russian invasion on February 24.
"I think that whoever started this war will be able to end it," Zelensky said, adding he was "not afraid" to meet the Russian leader. But he again stressed that Kyiv would abandon talks with Moscow if its troops in the besieged port city of Mariupol were killed.
Around 200 residents gathered at an evacuation meeting point announced by Kyiv in Mariupol on Saturday but they were "dispersed" by Russian forces, city official Petro Andryushchenko said on Telegram, adding: "The evacuation was thwarted."
He claimed others had been told to board buses headed to places controlled by Russia. The strategic city has been devastated by weeks of intense Russian bombardment.
Deputy Prime Minister Iryna Vereshchuk had said earlier that Ukraine would try again to evacuate women, children and the elderly from the city -- pivotal to Russia's war plans, and which the Kremlin claims to have "liberated".
Ukraine says hundreds of its forces and civilians are holed up inside a sprawling steel plant in Mariupol, and Kyiv has repeatedly called for a ceasefire to allow civilians to exit safely.
But on Saturday a Ukrainian presidential adviser, Oleksiy Arestovich, said Russian forces had resumed air strikes on the factory.
Six dead in Odessa
"Our defenders hold on regardless of the very difficult situation and even carry out counter raids," he said.
Further west, Russia said it had targeted a major depot stocking foreign weapons near Odessa on the Black Sea coast.
"Russian armed forces today disabled with high-precision and long-range missiles a logistics terminal at the military airfield near Odessa where a large batch of foreign weapons delivered by the United States and European countries were stored," the Russian defence ministry said in a statement.
Another strike on Odessa killed six people, including a three-month-old baby, Ukrainian officials said, upending the relative calm the city has enjoyed since the beginning of the war.
The country's emergency services said a missile struck a 15-storey residential building, sparking a fire that took 90 minutes to extinguish.
Odessa city hall said eight people were admitted to hospital.
"It was a terrifying night," said Yelena, with black bags under tear-reddened eyes in Ukraine's second city of Kharkiv, where residents said random Russian strikes could come at any hour, day or night.
"At 10:00 pm it all started, everything shook," she recalled. "There were two strikes, later there were more, we were no longer able to sleep and spent all night in the corridor.
The governor of Ukraine's eastern Kharkiv region, Oleg Sinegubov, said on Telegram that Ukranian forces had retaken three villages near the Russian border after "fierce battles" in which two people had been killed.
'Evacuate if you can'
In nearby Lugansk, governor Sergiy Gaiday said shelling was "round the clock" and urged people near the front to "evacuate if you have the chance".
The latest fighting comes a day after a senior Russian military officer said "the second phase of the special operation" had begun.
"One of the tasks of the Russian army is to establish full control over the Donbas and southern Ukraine," Major General Rustam Minnekaev said.
Russian forces, which withdrew from around Kyiv and the north of Ukraine after being frustrated in their attempts to take the capital, already occupy much of the eastern Donbas region and the south.
Minnekaev said the focus was to "provide a land corridor to Crimea," which Russia annexed in 2014, and towards a breakaway pro-Russian region of Moldova, Transnistria, where the general claimed Russian-speaking people were "being oppressed".
Ukrainian authorities have vowed to fight on and drive the Russian troops from their land, but they also sought an Easter pause.
"Unfortunately, Russia rejected the proposal to establish an Easter truce," Zelensky said earlier this week.
Ukrainian authorities urged those celebrating Orthodox Easter to follow religious services online and to respect curfews in place across the country.
"Support the defenders of Ukraine and stay at home for our security and yours," said Kyrylo Timoshenko from the president's office.
Roman Starovoit, the governor of Russia's region of Kursk, which borders Ukraine, said on Telegram that a Russian border post had been hit by Ukranian mortar fire, although there were no casualties.
Near the frontlines in eastern Ukraine, the hamlet of Lysychansk has largely turned into a ghost town but a small market is still operating, providing food and other supplies after the town's other market was bombed.
"This is going to end badly," said an elderly woman in line for vegetables, fearing a targeted strike by Russian forces similar to a deadly train station rocket attack in the nearby town of Kramatorsk on April 8 that killed at least 52 people.
Russia's change of strategic focus to southern and eastern Ukraine saw invading forces leave behind a trail of indiscriminate destruction and civilian bodies around Kyiv, including in the commuter town of Bucha.
A United Nations mission to Bucha documented "the unlawful killing, including by summary execution, of some 50 civilians there", the UN's Office of the High Commissioner for Human Rights said.
Russian forces had "indiscriminately shelled and bombed populated areas, killing civilians and wrecking hospitals, schools and other civilian infrastructure, actions that may amount to war crimes".