KYIV — With evacuations expanding to a second Russian border region, President Putin on Monday said that he will “kick the enemy out,” as Ukrainian military forces press deeper into Russian territory.

Mr. Putin said Kyiv’s purpose in entering Russia is to improve its negotiating position. He said Ukraine would receive a “worthy response” and added that he was confident Russia would reach all of its goals.  

“Losses have been mounting dramatically for the armed forces of Ukraine, particularly among the most capable units which the enemy has been sending to our border,” Mr. Putin said during a televised meeting with government officials. 

The governor of Russia’s Kursk region, Alexei Smirnov, said during the meeting with Mr. Putin that 121,000 residents of the area have evacuated. Mr. Smirnov added that Ukrainian troops control 28 villages in the region. 

“The situation remains difficult,” he said. He also told Mr. Putin that 2,000 Russian civilians are still in areas occupied by Ukrainian forces. “We don’t know anything about their fate,” he added.

In the meantime, residents from another Russian border region, Belgorod, have been asked to evacuate on Monday due to “enemy action,” Governor Vyacheslav Gladjov said via his Telegram channel. About 11,000 people are currently leaving Belgorod, which is situated next to the Kursk region. Mr. Gladjov also said the area was under missile alert and asked people to find shelter in basements. 

In a video on social media, Ukrainian soldiers posed in front of a clubhouse in the Poroz city of Belgorod, holding a Ukrainian flag. Other footage shows Ukrainian troops removing a Russian flag from a building in the Belgorod oblast, an archaic Soviet term for an administrative area. 

Ukrainian forces have advanced 18 miles into Russia’s territory since the incursion began last Tuesday. It has become the biggest assault on Russian soil since the start of the war.

According to an American think tank, the Institute for the Study of War, Kyiv’s army has continued to advance to the west and north of the Kursk oblast despite Russian sources saying that the situation in the region has stabilized and that Ukraine’s offensive has slowed. 

A video released Monday shows members of the Ukrainian military circulating through the central part of the town of Sudzha, in Kursk. The capture of Sudzha could mark a strategic boost for Kyiv because it is the junction for two major Russian gas pipelines, bringing energy supplies to the west from Siberia and Russia’s Arctic territories. 

Analyst Emil Kastehelmi, with a Finland-based organization that analyzes military movements during Russia’s war in Ukraine, Black Bird Group, wrote that Ukraine has not recently made significant progress as Russia has begun to counterattack. Yet, he added that Kyiv troops had solidified their positions in captured areas.

President Zelensky of Ukraine addressed the nation on Sunday and said that Russia has been launching missiles into Ukraine’s territory from the Kursk and Belgorod regions. 

“Since the beginning of the summer alone, almost 2,000 strikes have been made on our Sumy region from districts of the Kursk region,” he said. “Every one of these strikes deserves a fair response,” he added, as he urged Western allies to allow long-range strikes into Russia.

Mr. Zelensky also blamed Russia for starting a fire at Europe’s largest nuclear power plant. 

“As long as the Russian terrorists maintain control over the nuclear plant, the situation is not and cannot be normal,” Mr. Zelensky said Sunday. “Since the first day of its seizure, Russia has been using the Zaporizhzhia NPP only to blackmail Ukraine, all of Europe, and the world.” 

The Russian-installed governor of the region where the plant is situated, Yevgeny Balitsky, blamed Ukrainian shelling for the fire but asked people to remain calm, as there had been no leakage of radiation.

In the meantime, the two nations are fighting over control of another critical nuclear power plant in Russia’s Kursk. Local press reported Saturday that Russian forces have been deployed to defend the Kursk Nuclear Power Plant as Ukrainian troops extend their attack farther into Russia. 

The plant is at the city of Kurchatov and is 49 miles away from Ukrainian-held Sudzha. 

Yet, Mr. Kastehelmi wonders: “What can Ukraine achieve if it spends more manpower and equipment to take more border villages in the Kursk direction?” 

“A larger land area would, of course, have more weight in possible peace negotiations in the future. Even if Ukraine were to reach the Rylsk-Korenevo-Sudzha line, it would likely still be less valuable than many of the territories currently occupied by Russia in Ukraine,” he added.