The Russian-installed governor of Ukraine's southern Kherson region told residents on Thursday to take their children and flee, in one of the starkest signs yet that Moscow is losing its grip on territory it claims to have annexed.

In a video statement on Telegram, Vladimir Saldo publicly asked for Moscow's help transporting civilians to safer regions of Russia.

"Every day, the cities of Kherson region are subjected to missile attacks," Saldo said. "As such, the leadership of Kherson administration has decided to provide Kherson families with the option to travel to other regions of the Russian Federation to rest and study."

"We suggested that all residents of the Kherson region, if they wish, to protect themselves from the consequences of missile strikes, ... go to other regions," he said, advising people to "leave with their children."

Kherson is one of four partially occupied Ukrainian provinces that Russia claims to have annexed this month, and arguably the most strategically important. It controls both the only land route to the Crimea peninsula Russia seized in 2014, and the mouth of the Dnipro, the giant river that bisects Ukraine.

Since the start of October, Ukrainian forces have burst through Russia's front lines there in their biggest advance in the south since the war began. They have since been advancing rapidly along the west bank of the Dnipro, aiming to cut off thousands of Russian troops from supply lines and potential escape routes back across it.

Moments after Saldo's message, Saldo's deputy, Kirill Stremousov, issued a statement denying any plan to evacuate.

"There is and can be no evacuation in Kherson region," Stremousov said, adding in a written comment: "Nobody is planning to withdraw Russian troops from the Kherson region."

A flight of civilians from Kherson would be a major blow to Russia's claim to have annexed around 15% of Ukraine's territory this month and incorporated an area the size of Portugal into Russia.

Russia has concentrated many of its best-trained troops to defend its grip on the regional capital, Kherson, the only large Ukrainian city it has captured intact since its invasion in February, and its only foothold on the Dnipro's west bank. But that force can only be supplied across the river, which is several kilometers wide and has few crossings.

In the past two weeks Ukraine has recaptured a swath of territory, with its forces bearing down towards the 3 km-long (2 mile-long) Nova Kakhovka dam that provides one of the last river crossings.

Mykolaiv, the nearest big Ukrainian-held city to Kherson, came under massive Russian bombardment on Thursday, with civilian facilities hit, local officials said.

Regional governor Vitaly Kim said the top two floors of a five-story residential building were destroyed and the rest were under rubble. Video footage provided by state emergency services showed rescuers pulling out an 11-year-old boy who Kim said had spent six hours trapped under the rubble.

In the east, three Russian missiles exploded on Thursday morning near the central market in Kupiansk, a major railway junction city that Ukrainian forces recaptured during their big advance there in September.

The missiles destroyed shops, carpeting surrounding streets with glass shards, rubble, and twisted metal sheets.

Dmytro, who declined to give his last name, trudged up the debris-strewn steps of his shoe store to salvage whatever undamaged inventory he could from the devastated interior.

"Who knows? They consider it a military object," he said sarcastically when asked why he thought the Russians hit Kupiansk's commercial center.

Ukraine's gains in the south this month followed rapid gains in the east since September.

Russian President Vladimir Putin has responded to the battlefield setbacks with dramatic moves to escalate the conflict: proclaiming the annexation of territory, calling up hundreds of thousands of reservists and repeatedly threatening to use nuclear weapons to protect Russia.

This week, Putin launched the biggest air strikes since the start of the war, firing more than 100 cruise missiles mainly at electricity and heat infrastructure, with some landing in parks, busy roads and at tourist sites. Putin said the strikes were retaliation for a blast that damaged Russia's bridge to Crimea.

Although the nationwide air strikes have tapered since Tuesday, Russian forces have continued to strike towns and cities closer to the front.

On Thursday NATO allies meeting in Brussels unveiled plans to also jointly beef up Europe's air defenses with Patriot and other missile systems.

"We are living in threatening, dangerous times," said German Defence Minister Christine Lambrecht at a signing ceremony where Germany and more than a dozen European NATO members committed to jointly procuring weapons for a "European Sky Shield."

Moscow said more military aid for Kyiv made members of the U.S.-led military alliance "a direct party to the conflict" and said admitting Ukraine to NATO would trigger a global conflict.

Ukrainian President Volodymyr Zelenskiy told the Parliamentary Assembly of the Council of Europe on Thursday that Ukraine still had about only 10% of what it needed to protect itself against Russian air attacks.