Russian troops are withdrawing from around Ukraine's second-largest city after bombarding it for weeks, the Ukrainian military said Saturday (local time), as Kyiv and Moscow's forces engaged in a grinding battle for the country's eastern industrial heartland. 

Ukraine's general staff said the Russian forces were pulling back from the northeastern city of Kharkiv and focusing on guarding supply routes, while launching mortar, artillery and airstrikes in the eastern province of Donetsk in order to "deplete Ukrainian forces and destroy fortifications". 

Ukrainian Defence Minister Oleksii Reznikov said the country was "entering a new — long-term — phase of the war". 

Ukrainian President Volodymyr Zelenskyy, meanwhile, said Ukrainians were doing their "maximum" to drive out the invaders and that the outcome of the war would depend on support from Europe and other allies. 

Firefighters extinguish a fire following a Russian bombardment at a park in Kharkiv, Ukraine. Photo / AP 

"No one today can predict how long this war will last," Zelenskyy said in his nightly video address late Friday. 

In a show of support, a US Senate delegation led by Minority Leader Mitch McConnell met with the Ukrainian president Saturday in Kyiv. A video posted on Zelenskyy's Telegram account showed McConnell, who represents the state of Kentucky, and fellow Republican senators Susan Collins of Maine, John Barrasso of Wyoming and John Cornyn of Texas greeting him. 

Their trip came after Kentucky's other senator, Rand Paul, blocked until next week Senate approval of an additional US$40 billion to help Ukraine and its allies withstand Russia's three-month-old invasion. 

After failing to capture Kyiv following the February 24 invasion, Russian President Vladimir Putin has shifted his focus eastward to the Donbas, an industrial region where Ukraine has battled Moscow-backed separatists since 2014. 

Russia's offensive aims to encircle Ukraine's most experienced and best-equipped troops, who are deployed in the east, and to seize parts of the Donbas that remain in Ukraine's control. 

Airstrikes and artillery barrages make it extremely dangerous for journalists to move around in the east, hindering efforts to get a full picture of the fighting. But it appears to be a back-and-forth slog without major breakthroughs on either side. 

Russia has captured some Donbas villages and towns, including Rubizhne, which had a prewar population of around 55,000. 

Zelenskyy said Ukraine's forces have also made progress in the east, retaking six towns or villages in the past day. 

Kharkiv, which is near the Russian border and only 80hm southwest of the Russian city of Belgorod, has undergone weeks of intense shelling. The largely Russian-speaking city with a prewar population of 1.4 million was a key Russian military objective earlier in the war, when Moscow hoped to capture and hold major cities. 

Ukraine "appears to have won the Battle of Kharkiv," the Institute for the Study of War, a Washington-based think tank, said. "Ukrainian forces prevented Russian troops from encircling, let alone seizing Kharkiv, and then expelled them from around the city, as they did to Russian forces attempting to seize Kyiv." 

Regional Gov. Oleh Sinegubov said via the Telegram messaging app that there had been no shelling attacks on Kharkiv in the past day. 

He added that Ukraine launched a counteroffensive near Izyum, a city 125km south of Kharkiv that has been held by Russia since at least the beginning of April. 

Dozens of destroyed or damaged Russian armoured vehicles on both banks of Siverskyi Donets River in eastern Ukraine. Photo / AP 

Fighting was fierce on the Siversky Donets River near the city of Severodonetsk, where Ukraine has launched counterattacks but failed to halt Russia's advance, said Oleh Zhdanov, an independent Ukrainian military analyst. 

"The fate of a large portion of the Ukrainian army is being decided — there are about 40,000 Ukrainian soldiers," he said. 

However, Russian forces suffered heavy losses in a Ukrainian attack that destroyed a pontoon bridge they were using to try to cross the same river in the town of Bilohorivka, Ukrainian and British officials said. 

Britain's defence ministry said Russia lost "significant armoured maneuver elements" of at least one battalion tactical group in the attack. A Russian battalion tactical group consists of about 1,000 troops. 

The ministry said the risky river crossing was a sign of "the pressure the Russian commanders are under to make progress in their operations in eastern Ukraine". 

Dozens of destroyed or damaged Russian armoured vehicles on both banks of Siverskyi Donets River in eastern Ukraine. Photo / AP 

In his video address, Zelenskyy warned that the war was causing a food crisis around the world as Russian blockades Ukrainian grain from leaving port. 

The Group of Seven leading economies echoed that, saying Saturday that "Russia's war of aggression has generated one of the most severe food and energy crises in recent history, which now threatens those most vulnerable across the globe". 

Putin launched the war in Ukraine aiming to thwart Nato's expansion in Eastern Europe. 

But the invasion has other countries along Russia's flank worried they could be next, and this week the president and prime minister of Finland said they favour seeking Nato membership. Officials in Sweden are expected to announce a decision Sunday on whether to apply to join the Western military alliance. 

In a phone call Saturday, Putin told Finnish President Sauli Niinisto that there are no threats to Finland's security and joining Nato would be an "error" and "negatively affect Russian-Finnish relations". 

The Kremlin said the two leaders had a "frank exchange of views". 

Niinisto said the discussion "was straightforward and unambiguous and was held without exaggeration. Avoiding tensions was considered important." 

Russia's response to the moves by Finland and Sweden has so far been muted, though Deputy Foreign Minister Alexander Grushko said Saturday that their accession to Nato would heighten security tensions in the Arctic, "turning it into an arena of military competition". 

Russian energy group Inter RAO suspended deliveries of electricity to Finland on Saturday, according to a statement from the Finnish national electrical grid operator. But only around 10 per cent of Finland's electricity comes from Russia, and authorities did not expect shortages. 

The Nordic nations' potential bids were thrown into question Friday when Turkish President Recep Tayyip Erdogan said his country is "not of a favourable opinion". 

US Secretary of State Antony Blinken is scheduled to meet his Nato counterparts, including the Turkish foreign minister, this weekend in Germany. 

 - Oleksandr Stashevski and David Keyton, AP