A 40-mile convoy of Russian tanks and other vehicles threatened Ukraine’s capital on Tuesday, the sixth day of the war. But even as Russia intensified shelling of the country’s second-largest city, the Kremlin has found itself increasingly isolated by tough economic sanctions that have sent its currency plummeting.

After a first, five-hour session of talks between Ukraine and Russia yielded no stop in the fighting, both sides agreed to another meeting in coming days. Ukraine’s embattled president, however, said he believed the stepped-up shelling was designed to force him into concessions.

“I believe Russia is trying to put pressure (on Ukraine) with this simple method,” Ukrainian President Volodymyr Zelenskyy said late on Monday in a video address. He did not offer details of the talks that took place on Monday, but he said Kyiv was not prepared to make concessions “when one side is hitting another with rocket artillery”.

“For the enemy, Kyiv is the key target,” Zelenskyy said. “We did not let them break the defence of the capital, and they send saboteurs to us ... We will neutralise them all.”

Zelenskyy said Russia, which calls its actions in Ukraine a “special operation”, was targeting a thermal power plant providing electricity to Kyiv, a city of three million people.

Six days into the invasion, the Russian military’s movements have been stalled by fierce resistance on the ground and a surprising inability to dominate the airspace. Many Ukrainian civilians, meanwhile, spent another night huddled in shelters, basements or corridors.

“I sit and pray for these negotiations to end successfully, so that they reach an agreement to end the slaughter,” said Alexandra Mikhailova, weeping as she clutched her cat in a shelter in Mariupol. Around her, parents tried to console children and keep them warm.

Ukrainian officials also reported a Russian bombardment of Kharkiv, the country's second-largest city, had killed dozens of civilians. It was not possible to independently verify the casualty figures.

“Barbaric rocket attacks and MLRS (multiple launch rocket systems) of peaceful cities are evidence that they are no longer able to fight armed Ukrainians,” Ukrainian Defence Minister Oleksii Reznikov said on Facebook.

The Kremlin has twice in as many days raised the specter of nuclear war and put on high alert an arsenal that includes intercontinental ballistic missiles and long-range bombers. Stepping up his rhetoric, President Vladimir Putin denounced the United States and its allies as an “empire of lies”.

Russian army reaches Kherson

The Russian army on Tuesday reached the southern Ukrainian city of Kherson, near Moscow-controlled Crimea, and is setting up checkpoints on its outskirts, the city's mayor said.

“The Russian army is setting up checkpoints at the entrances of Kherson,” mayor Igor Kolykhayev said on Facebook. “Kherson has been and will stay Ukrainian,” he said.

Videos on local media in Kherson showed the Russian army entering the city.

“Today, I am responsible for the life of our city and provide protection in the way that our capabilities allow,” Kholykhayev said.

He asked people not to leave their homes outside curfew hours.

“The greatest value this city has is your lives. This is not a fight, this is a war,” he said.

Kherson has a population of around 280,000 and lies north of the Crimea peninsula, annexed by Moscow in 2014.

Russia claimed to have besieged the city on Sunday.

The Russian army also says it controls the port city of Berdyansk, which lies north-west of Crimea on the Azov Sea.

Fighting raged in other towns and cities. The strategic port city of Mariupol, on the Sea of Azov, is “hanging on”, said Zelenskyy adviser Oleksiy Arestovich. An oil depot was reported bombed in the eastern city of Sumy.

Russian artillery hit a military base in Okhtyrka, a city between Kharkiv and Kyiv, and more than 70 Ukrainian soldiers were killed, the head of the region wrote on Telegram. Dmytro Zhyvytskyy posted photographs of the charred shell of a four-story building and rescuers searching rubble.

In a later Facebook post, he said many Russian soldiers and some local residents also were killed during the fighting on Sunday. The report could not immediately be confirmed.

Ceasefire talks

Ceasefire talks between Russia and its southern neighbour held on Monday failed to reach a breakthrough and negotiators have not said when a new round would take place.

A top Putin aide and head of the Russian delegation, Vladimir Medinsky, said that during the first talks held between the sides since the invasion, the envoys “found certain points on which common positions could be foreseen”. He said they agreed to continue the discussions in the days ahead.

Meanwhile, an embattled Ukraine moved to solidify its ties to the West by applying to join the European Union — a largely symbolic move for now, but one that won’t sit well with Putin, who has long accused the United States of trying to pull Ukraine out of Moscow’s orbit.

Putin faces mounting international pressure for last week launching the biggest assault on a European state since World War Two, and the systemic impact of Western sanctions led to a near 30 per cent collapse in the rouble on Monday before central bank intervention rescued the currency from its lows.

Turkey shuts straits to warships

Nato ally Turkey delivered another blow to Moscow on Monday by warning warring countries not to send warships through its Bosphorus and Dardanelles Straits that separate the Black Sea from the Mediterranean, effectively bottling up Russia's Black Sea Fleet.

Washington has ruled out sending troops to fight Russia or enforcing a no-fly zone as requested by Ukraine, fearing an escalation between the world's top two nuclear powers.

But, the United States and its allies have instead promised military aid to Kyiv.

Human rights groups and Ukraine's ambassador to the United States accused Russia of using cluster bombs and vacuum bombs. The United States said it had no confirmation of their use.

Public health experts say Ukraine is running low on critical medical supplies and fears of a wider public health crisis are growing as people flee their homes and health services and supplies are interrupted.

Russia says its actions are not designed to occupy territory but to destroy Ukraine's military capabilities and capture what it regards as dangerous nationalists.