KYIV—Moscow battled to regain control of its border region as Ukrainian forces pushed deeper into Russian territory four days after a surprise incursion.

As Russia struggled to push Ukrainian soldiers back, Kyiv’s forces struck at least one Russian military column and claimed to be in control of a town in the lightly defended Kursk region.

Ukraine carried out cross-border raids earlier in the conflict, but the scale of the latest assault is significantly larger. Ukraine now faces a decision on whether to commit additional forces to a promising but risky incursion given the pressure Russian troops are exerting on Kyiv’s threadbare front lines in the east.

Ukrainian officials haven’t commented directly on the incursion, but President Volodymyr Zelensky said Friday that he had received “productive” reports from his military chief, Col. Gen. Oleksandr Syrskiy.

“Russia brought the war to our land, and it should feel what it has done,” he said.

Russia’s defense ministry said its forces “continue to repel an attempted invasion by the Armed Forces of Ukraine into the territory of the Russian Federation.” Earlier this week, the country’s top military commander, Gen. Valery Gerasimov , reported to President Vladimir Putin that the advance had been halted.

Video posted on social media and verified by Storyful, owned by The Wall Street Journal’s parent company News Corp , showed a convoy of more than a dozen burnt-out Russian military vehicles—some containing corpses—on a highway in the Kursk region. In another video, Ukrainian soldiers claimed to be in control of the town of Sudzha, about 6 miles inside Russia.

The Russian defense ministry said its forces had destroyed five Stryker armored fighting vehicles operated by Kyiv’s forces. The mayor of Sudzha denied the town was under Ukrainian control in comments to Russian state media and said an evacuation of local residents was under way.

In videos circulated on social media, displaced civilians pleaded with Putin to come to their aid. “Putin, please come here and help protect us,” screamed one woman.

U.S. National Security Council spokesman John Kirby indicated Friday the U.S. still doesn’t know what Ukraine intends to accomplish with its incursion into Russia. “We’re in touch with our Ukrainian counterparts, and we are working to gain a better understanding of what they’re doing, what their goals are, what their strategy is,” he said. The U.S. previously said it didn’t receive advance notice of the operation, which is normal for a tactical move.

The cross-border attack comes at a time when Ukrainian forces are under intense pressure along the front line in the east of the country. Russia is advancing toward the critical Ukrainian logistical hub of Pokrovsk and mounting relentless attacks on the nearby city of Chasiv Yar, located on strategically important heights in the eastern Donbas region.

By pushing into Russian territory, Ukraine may be hoping to draw resources away from other parts of the front.

The number of troops involved in the raid is small compared with the hundreds of thousands deployed on each side within Ukraine.

As fighting raged in Kursk, Rafael Grossi , director general of the International Atomic Energy Agency, voiced concern about reports of fighting in the region where a nuclear power station is located. Grossi said he was monitoring the situation and was in contact with the relevant authorities of both countries.

“At this juncture, I would like to appeal to all sides to exercise maximum restraint in order to avoid a nuclear accident with the potential for serious radiological consequences,” he said.

Alex Ward, Matthew Luxmoore and Laurence Norman also contributed to this article.

Write to Isabel Coles at isabel.coles@wsj.com