Russian forces continued their offensive in eastern Ukraine, closing in on strategic towns and cities and shelling Kharkiv after Ukrainian President Volodymyr Zelenskiy visited the northeastern city in his first trip outside Kyiv since the beginning of the full-scale Russian invasion.

Russian forces increased the pressure on Syevyerodonetsk and Lysychansk in eastern Ukraine after claiming to take the nearby city of Lyman, with Ukrainian officials saying defenders were holding out under desperate conditions in those strategic areas.

Luhansk regional governor Serhiy Hayday said the situation in Lysychansk had become significantly worse in recent hours and that “there are fatalities and wounded people.”

In Syevyerodonetsk, the Ukrainian General Staff said Russian forces had carried out an assault operation while Hayday said street-by-street fighting was raging in the city situated hear the banks of the Donets River.

Zelenskiy on May 29 paid a surprise visit to the northeastern city of Kharkiv, venturing out of the capital, Kyiv, for the first time since Russia’s unprovoked invasion began on February 24.

He visited with soldiers on frontline positions and held a session with local officials, including Kharkiv Mayor Ihor Terekhov, regional administrator Oleh Synyehubov, and members of the military command.

"You risk your lives for us all and for our country," Zelenskiy’s office quoted the president as telling troops fighting there.

The presidential office posted a video on the Telegram messaging app of Zelenskiy in a bullet-proof vest as he toured areas devastated by Russian shelling in and around the city over recent weeks.

Reuters reported that hours after Zelenskiy’s visit, several blasts were heard in the city and that a large plume of smoke was visible northeast of the central area.

The State Emergency Service of Ukraine said Russian shelling had caused fires around Kharkiv, the country's second-largest city. Russia has kept up a bombardment of Kharkiv after Ukrainian fighters pushed its forces back from positions near the city several weeks ago.

Russian forces have turned their efforts to the eastern part of Ukraine -- specifically the Donbas region -- after stiff resistance prevented them from taking Kyiv at the beginning of the invasion.

In comments to French television, Russian Foreign Minister Sergei Lavrov on May 29 said what he called the liberation of Ukraine's Donbas region was an unconditional priority for Moscow. Other Ukrainian territories should decide their own future, he said.

The Ukrainian president’s office said Synyehubov told Zelenskiy that Ukrainian forces had made some progress in their counterattack against Russia’s latest offensive.

"But we are not yet able to fully inspect some of the liberated settlements, as shelling continues, or to conduct full-fledged demining and begin rebuilding critical infrastructure,” Synyehubov was quoted as saying.

He added that Russian forces had damaged 2,229 high-rise buildings, of which 225 were completely ruined, in the Kharkiv region. He said the northern and eastern districts of Kharkiv had suffered the most damage, with more than 30 percent of the housing stock destroyed.

The damage figures -- as is common from both sides in the conflict -- could not immediately be verified.

Earlier, in his late-night video address on May 28, Zelenskiy expressed hope that allies will provide needed weapons as Ukrainian forces try to halt advancing invading Russian forces in the east.

Zelenskiy said that he expected good news on weapons deliveries this week, without giving details.

Zelenskiy said the military situation in the Donbas region was complicated, adding that defenses were holding up in a number of places, including Syevyerodonetsk and Lysychansk, the last major areas under Ukrainian control in the Luhansk region.

"It's indescribably difficult there. And I am grateful to all those who withstood this onslaught," he said.

Russian invading forces are reported to have made gains in recent days in eastern Ukraine’s Donbas, comprising the Luhansk and Donetsk regions.

Fighting for control of Syevyerodonetsk continues with Russian forces conducting assault operations on May 28, the General Staff of Ukraine's Armed Forces said on May 29.

"With the use of artillery, Russian forces carried out assault operations in the area of the city of Syevyerodonetsk," the General Staff said in a statement on its Facebook page.

Hayday, the governor of Luhansk, said Russian forces had dug in at the Myr hotel in northern Syevyerodonetsk.

He said the bulk of Russian forces were unable to advance into the city center and were taking casualties, but he also said Ukrainian troops were not currently able to "push them out of the hotel."

Zelenskiy vowed to maintain Ukrainian resistance in the important city.

"Some 90 percent of buildings are damaged. More than two-thirds of the city's housing stock has been completely destroyed. There is no telecommunication. There is constant shelling," Zelenskiy said.

"Capturing Syevyerodonetsk is a fundamental task for the occupiers...We are doing all we can to hold this advance," he added.

Russian artillery was also pounding the Lysychansk-Bakhmut road, which Russia must take to close a pincer movement and encircle Ukrainian forces, and police said there was significant destruction in Lysychansk.

Ukrainian officials said on May 29 that Russian forces had been pushed back from Bakhmut, although battlefield details could not immediately be verified.

The British Defense Ministry said in its daily intelligence report on May 28 that if Russia succeeded in taking over those areas, the Kremlin would likely view it as a substantive political achievement, which it could use to justify its invasion to the Russian people.

In his late-night address, Zelenskiy said Ukraine was approaching the point where it would outnumber the Russians both technologically and in terms of its ability to strike.

Ukrainian Defense Minister Oleksiy Reznikov said Ukraine has started receiving Harpoon anti-ship missiles from Denmark and self-propelled howitzers from the United States.

"The coastal defense of our country will not only be strengthened by Harpoon missiles -- they will be used by trained Ukrainian teams," Reznikov wrote on his Facebook page on May 28.

He said Harpoon shore-to-ship missiles would be operated alongside Ukrainian Neptune missiles in the defense of the country's coast including the southern port of Odesa.

Last month, the Moskva, the flagship of Russia's Black Sea fleet, sank after what Ukraine says was an anti-ship missile attack. Moscow says a fire sparked an ammunition blast.