Ukrainian officials said the Kharkiv attacks had killed civilians, including children.
The European Union placed new sanctions on Russian oligarchs and officials, and some of its members urged the bloc to begin talks on Ukrainian accession.
President Volodymyr Zelenskiy signed a letter formally requesting EU membership, an emphatic statement of commitment to Western values.
But Putin showed no sign of reconsidering the invasion he unleashed on Russia's neighbour last Thursday in an attempt to pull it firmly back under Moscow's influence and redraw Europe's security map.
He dismissed the West as an "empire of lies" and reacted to the new sanctions with moves to shore up Russia's crumbling rouble currency.
Kharkiv in Ukraine's northeast has become a major battleground.
Regional administration chief Oleg Synegubov said Russian artillery had pounded residential districts. At least 11 people were killed, he said.
"This is happening in the daytime, when people have gone out to the pharmacy, for groceries, or for drinking water. It's a crime," he said.
Kharkiv's mayor, Igor Terekhov, said four people had died after emerging from a bomb shelter to collect water, and a family with three children had burned to death in a car.
Moscow's United Nations ambassador, speaking in New York, said the Russian army did not pose a threat to civilians.
Images from the US satellite company Maxar appeared to show a Russian military convoy stretching over 27km and moving closer to the capital Kyiv.
Fighting occurred throughout Sunday night around the port city of Mariupol, said the head of the Donetsk regional administration, Pavlo Kyrylenko.
Russian forces seized two small cities in southeastern Ukraine and the area around a nuclear power plant, according to the Interfax news agency.
Talks between the two sides were held on the border with strong Russian ally Belarus, which has been used as a launch pad for invading Russian troops.
The meeting ended with officials heading back to capitals for further consultations.
"The Russian side, unfortunately, still has a very biased view of the destructive processes it has launched," Ukrainian presidential adviser Mykhailo Podolyak tweeted.
Russian delegation head Vladimir Medinsky told reporters: "The most important thing is that we agreed to continue negotiating."
Russia calls its actions in Ukraine a "special operation" that it says is not designed to occupy territory but to destroy its southern neighbour's military capabilities and capture what it regards as dangerous nationalists.
The Western-led response has been emphatic, with sanctions that effectively cut off Moscow's financial institutions from Western markets.
The rouble plunged 32 per cent against the US dollar before recouping about half of its losses, and Russia's central bank cranked up its key interest rate to 20 per cent from 9.5 per cent.
In Brussels, European Union foreign policy chief Josep Borrell said EU sanctions would have a cost for Europe "but we have to be ready to pay the price, or we will have to pay a much higher price in the future".
The invasion has brought relations between the United States and Russia, the world's two biggest nuclear powers, to their worst point in years.
The United States expelled 12 Russian diplomats at the United Nations, citing national security concerns. Russia described the move as "hostile".
UN human rights chief Michelle Bachelet said at least 102 civilians in Ukraine had been killed since Thursday but the real figure could be "considerably higher".
Ukraine's health ministry said on Sunday 352 civilians, including 14 children, had been killed since the beginning of the invasion.
More than half a million people have fled to neighbouring countries, according to the UN refugee agency.
Partners in the US-led NATO defence alliance were providing Ukraine with air-defence missiles and anti-tank weapons, NATO chief Jens Stoltenberg said.
The Kremlin accused the EU of hostile behaviour, saying weapons supplies to Ukraine were destabilising and proved Russia was right in its efforts to demilitarise its neighbour.