As Ukraine ramps up its offensive to retake Russian-occupied territory in the country's south, its forces have regained four villages and broken through Russian defenses at multiple points of the frontline in the Kherson region, according to Ukrainian officials.

Ukraine's President Volodymyr Zelensky also issued a warning to Russian forces during his evening address Monday, saying it was time for them to "run away, go home," and that "the occupiers must know we will chase them to the border."

Yet one of his advisers cautioned that the operation will be a slow one, that would "grind the enemy."

"This process will not be very fast," Oleksiy Arestovych, adviser to the head of the office of the President of Ukraine, said in a statement posted on Telegram late Monday, "but will end with the installation of the Ukrainian flag over all the settlements of Ukraine."

"[The Ukrainian Armed Forces] have broken through the frontline in several sectors," Arestovych said in a video interview posted on YouTube, adding that Ukrainian forces were shelling ferry crossing points that Moscow is using to resupply Russian-occupied territory over the Dnipro River in the Kherson region.

"We suppress their attempts to supply their groups on the west bank [of the Dnipro]," he said. "We strike at reserves, including reserves that are on the east bank and are trying to cross." 

Earlier Monday, a Ukrainian military source told CNN that Ukrainian troops had taken four villages -- Novodmytrivka, Arkhanhel's'ke, Tomyna Balka and Pravdyne -- back from Russian occupation, adding that the main target was the city of Kherson.

"The operation began at night with massive shelling of Russian positions and the rear," the source, who is not being named for security reasons, told CNN.

Russia's defense ministry acknowledged Kyiv's operation but claimed that Ukrainian troops had "suffered heavy losses" and "failed miserably" in their "attempted" offensive. CNN has not been able to verify both Russian and Ukrainian claims.

The Kherson region, crucial to controlling Ukraine's southern coast and access to the Black Sea, was one of the first areas to be occupied by Russian forces when they invaded more than six months ago.

In recent weeks, Ukrainian forces have been making advances around Kherson, including the targeting of critical bridges to disrupt Russian supply routes.

Ukrainian forces also launched an attack at the Russian-held town of Nova Kakhovka on Monday, knocking out its electricity and water supply, according to Russian state media RIA Novosti, who cited a Russian-appointed local authority.

RIA quoted the head of civil-military administration in the town, Vladimir Leontyev, as saying the consequences of the strike "will probably be terrible."

"You can't leave now. There was a lot of shelling, something exploded. There were strikes in the city and near the hydroelectric plant. We can see the fire in both places," Leontyev added.