President Volodymyr Zelenskyy said late on Tuesday that his country was gradually pushing the Russian troops away from the key city, which has been under sustained attack since the start of the war.
The counteroffensive saw Ukrainian fighters drive the Russians out of four villages to the northeast of Kharkiv, according to the Ukrainian General Staff.
US-based think tank, The Institute of War, cited a Russian source who claimed Ukrainian fighters managed to advance to within 10km (six miles) of the Russian border.
Despite the Ukrainian advancements, Russian aircrafts twice launched unguided missiles on Tuesday at the Sumy area northeast of Kharkiv, according to the Ukrainian border guard service.
Regional governor Oleh Synehubov announced on Tuesday that the bodies of 44 civilians were found in the rubble of a five-story building that collapsed in March in Izyum, about 75 miles from Kharkiv.
Other key developments:
"No immediate chances of a peace agreement" in Ukraine, says top UN chief
Russia-installed official in Ukraine's Kherson region says he will ask Vladimir Putin to annex the region
Ukraine's natural gas pipeline operator stops Russian shipments through key hub in east
Ukraine's targeting of Russian troops on Snake Island helping block Moscow's attempts to expand its force in the Black Sea, says UK
US pledges $40 billion (£32,397,000) in Ukraine aid
At least four civilians were also killed while trying to evacuate from the frontline village - the convoy of 15 cars had been trying to evacuate to Kharkiv near Staryi Saltiv village, said Kharkiv's regional prosecutor's office.
One member of the Ukrainian military claimed the convoy was probably shot with an "automatic gun" and that one car had been hit by a cannon from a "tank or an infantry fighting vehicle". The wreckage was found on Friday.
Last month, Kharkiv's Governor Synehubov said there had not been a single day without strikes since the start of the invasion, forcing residents to shelter in dark, cold basements.
For expert analysis and insight on the biggest stories listen to our podcast to find out What You Need To Know
The latest developments came as the United Nations' top chief said he believed there was "no immediate chances of a peace agreement" or any "immediate chances of a global ceasefire" in Ukraine.
Secretary-General Antonio Guterres said on Wednesday that the UN was instead immediately focusing its efforts in evacuations and delivering aid to help the "dramatic humanitarian situation" in areas such as the battered city of Mariupol and the Azovstal steel plant.
"I hope nothing lasts forever. This war will not last forever," Mr Guterres told a press conference in Vienna.
"There will be a moment in which the peace negotiations will be on the table... But it's not in the immediate horizon. But one thing I can tell you: we will never give up."
The wives of two Ukrainian soldiers defending the Mariupol steelworks met with Pope Francis on Wednesday, urging him to intervene to arrange for a third-party evacuation of the troops before Russian soldiers capture or kill them.
One of them wept as she told Francis: “You are our last hope. We hope you can save their lives. Please don’t let them die."
Russian forces on Wednesday continued to pound the Azovstal steelworks, the last pocket of resistance of Ukrainian forces in the port city of Mariupol where an estimated 2,000 soldiers and 100 civilians are holed up, said the Azov Batallion.
It said in a statement that over the past 24 hours the Russians carried out 38 airstrikes "on the territory” of Azovstal, including with the use of strategic bombers.
Elsewhere, a Russia-installed official in Ukraine’s Kherson area reportedly said local officials will ask Russian President Vladimir Putin to annex the region.
Deputy head, Kirill Stremousov, told reporters on Wednesday that there are no plans to create a self-proclaimed “Kherson People’s Republic,” akin to the ones in Donetsk and Luhansk regions of Ukraine - but that there are plans to ask Putin to annex it.
Taking control of Kherson, a city in southern Ukraine, and much of the surrounding region early on the war has been arguably Russia’s most significant gain in the war.
“The city of Kherson is Russia,” Stremousov was quoted as saying by the state RIA Novosti news agency.
“There will be no referendums. It will be a decree based on an appeal from the Kherson regional leadership to the Russian president, and there will be a request to include the region into a proper region of the Russian Federation.”
However, in apparent further success for Ukraine, the British military says Ukraine’s targeting of Russian forces on Snake Island in the Black Sea is helping disrupt Moscow’s attempts to expand its influence in the Black Sea.
The Ministry of Defence said on Twitter: “Ukraine has successfully struck Russian air defences and resupply vessels with Bayraktar drones. Russia’s resupply vessels have minimum protection in the western Black Sea, following the Russian Navy’s retreat to Crimea after the loss of the Moskva.”
It came as President Joe Biden approved a fresh $40 billion Ukraine aid package, signalling a magnified, bipartisan commitment to thwart Putin’s now three-month-old invasion.
Meanwhile, Russian missiles pummelled the vital port of Odesa, apparently as part of efforts to disrupt supply lines and weapons shipments critical to Kyiv’s defence, said Ukrainian officials.
Ukraine’s natural gas pipeline operator has also stopped Russian shipments through a key hub in the east of the country. Wednesday's move was the first time natural gas supply has been affected by the war that began in February.
Experts say it could force Russia to shift flows of its gas through territory controlled by Ukraine to reach its clients in Europe.
Russia’s state energy giant Gazprom initially said it couldn’t reroute the gas, though preliminary flow data suggested higher rates moving through a second station in Ukrainian-controlled territory.
The pipeline operator said Russian shipments through its Novopskov hub, in an area controlled by Moscow-backed separatists, would be cut because of interference from “occupying forces,” including the apparent siphoning of gas.
The Ukrainian pipeline operator said the hub handles about a third of Russian gas passing through Ukraine to Western Europe. Russia’s state-owned natural gas giant Gazprom put the figure at about a quarter.
In his nightly address, President Zelenskyy also paid tribute to Leonid Kravchuk, the first president of an independent Ukraine, who died on Tuesday at 88.
He said Kravchuk showed courage and knew how to get the country to listen to him.
That was particularly important in “crisis moments, when the future of the whole country may depend on the courage of one man,” said Mr Zelenskyy, whose own communication skills and decision to remain in Kyiv when it came under Russian attack helped make him a strong wartime leader.