Ukraine urges civilians to flee the occupied south ahead of a promised counteroffensive.

A top Ukrainian government official has made an urgent plea for hundreds of thousands of people living in Russian-occupied parts of southern Ukraine to evacuate in advance of a potential Ukrainian counteroffensive, working to prepare the public for a bloody struggle on another front even as Russia makes steady gains in fierce and costly battles in the east.

In trying to take back territory in the south, Ukrainian officials are facing deep challenges. Russia has been dug in for months in parts of the region, complicating evacuation routes for civilians and forcing Kyiv to decide how much damage it is willing to inflict on towns and cities that — even if an eventual counteroffensive is successful — it would have to rebuild.

“Please, go, because our Army will definitely de-occupy these lands,” said Iryna Vereshchuk, Ukraine’s deputy prime minister. In one sign of the dilemmas facing Ukraine, she indicated that civilians might first have to flee to Crimea, which was seized by Russia in 2014 and has been a key staging ground for Moscow’s invasion.

Russian forces have intensified searches at checkpoints in the region and are preventing civilians from traveling to Ukrainian-controlled areas. The safest escape route for many of the 500,000 people still estimated to be living in the Kherson region may be to head south to Crimea, she said.

“We know that today it is almost the only humanitarian corridor available, if it can be called that, that can be used to leave,” Ms. Vereshchuk said at a news conference on Monday. “So, if possible, get out of there, especially if you have children.”

From Crimea, she urged people to make their way to another country where they can reach out to the local Ukrainian consulate. She did not elaborate on how Ukrainians would be able to leave from Crimea.

While it is not clear when or if Ukraine will launch a broad offensive, it has been staging limited counterattacks that have put the Russians on the defensive. Ukrainians are now battling Russian forces trying to hold defensive lines less than 12 miles from the city of Kherson — the only regional capital to fall to Russian forces since the Feb. 24 invasion.

The Russians have used indiscriminate, heavy bombardments to level whole towns and villages before infantry troops move in to claim the ruins. But like other towns and cities that fell quickly in the first days of the war, Kherson was spared the kind of widespread destruction that has defined the Russian advance.

Now the Ukrainians are wrestling with complicated calculations in the fight to reclaim lost lands.

Ukrainian officials have stressed how important it is to keep the local population on their side — a task that may be complicated after months of Russian occupation and propaganda. The Russian Ministry of Defense said on Tuesday that it had repaired the last of seven television towers in the Kherson region to broadcast Russian channels.

The Ukrainian military also hopes to use local residents behind enemy lines as a force multiplier by staging sabotage operations, scouting enemy targets and generally creating a hostile environment for occupation forces.

A picture taken during a media tour organized by the Russian Army shows a Russian servicemen standing guard near the Dnipro River in Kherson last month. Russian forces have been in control of the city since the early days of the war. Credit…Sergei Ilnitsky/EPA, via Shutterstock

That could include attacks like one over the weekend on three Russian soldiers as they sat at a waterfront cafe in Kherson. Two were killed and a third had to be transported to a hospital in Crimea for treatment after they came under attack by an unidentified gunman, according to the Ukrainian military southern command.

Ukraine would also need to assess how much damage it is willing to inflict in any effort to retake Kherson, since, if it succeeds, it would fall on Kyiv to rebuild. “If the city is demolished, then why liberate it?” asked Oleksandr Samoilenko, a Kherson official.

But the Russians have had months to fortify their positions.

“It will be very, very difficult to open a humanitarian corridor when there are children there,” Ms. Vereshchuk said. “It was difficult in Mariupol, and in the Kherson region it will be even harder.”

Military analysts have cautioned that it could most likely be several weeks before the Ukrainians get more powerful weapons and ammunition needed to wage a wide offensive in the south. But Ukrainian forces continued to set the stage for a broad assault.

The Ukrainian military’s southern command said on Tuesday that Russia was being forced to “resort to desperate means” to hold onto its outer defensive ring, including launching a “massive missile strike.”

There were no reported casualties and the region’s air defense systems limited the damage, according to the military.