Near the city of Lysychansk, Russia deployed more troops to try to capture the village of Bilohorivka, Ukraine's governor of the region said, while a commander in another heavily fought-over settlement described an intensifying Russian air offensive.
"They are bringing in more and more reserves," around Bilohorivka to try to capture the village, Luhansk governor Serhiy Haiday told Ukrainian television.
"There are constant attacks."
In the settlement of Bakhmut and other parts of the Donetsk region that neighbours Luhansk, the assault killed nine civilians, the regional governor said.
Ukrainian forces countered with barrages from rocket launchers, Reuters witnessed.
Fighting was under way along the entire line of demarcation in Donetsk, with the frontline town of Avdiivka shelled by Russian tanks on Thursday morning, said Tatiana Ignatchenko, a spokeswoman for the Donetsk regional administration.
Putin made clear on Wednesday that expanding Russia's borders was a key goal of the war, in contrast to stated aims at the start of the February 24 invasion he calls a "special military operation" when he said Moscow's plans did not include the occupation of Ukrainian land.
Putin said Russia had already achieved a "significant result" with the acquisition of "new territories" in Ukraine - a reference to the annexation of four partly occupied regions in September that Kyiv and most members of the United Nations condemned as illegal.
Warning that the war could be lengthy, Putin said Russia had made the Sea of Azov its "internal sea", now bounded by Russia and Russian-controlled territory in southern Ukraine including Crimea.
He said that had been an aspiration of Peter the Great - the 17th and 18th-century warrior tsar to whom he has compared himself in the past.
Kyiv's forces have in recent weeks pushed Russia back from swathes of land it occupied, including a major city, Kherson.
President Volodymyr Zelenskiy says his troops will eventually drive Russia from all the captured territory, including the annexed Crimea peninsular that sits between the Black Sea and the Sea of Azov.
On Thursday, Russian naval forces shot down a Ukrainian drone over the Black Sea, said the Russian-installed governor of Sevastopol, an important port and the largest city in Crimea.
Earlier this week, twin strikes on air bases deep inside Russian territory dealt Moscow a major reputational blow and raised questions about why its defences failed, as attention turned to the use of drones in the war between neighbours.
In the Russian-controlled Ukrainian city of Melitopol, Zaporizhzhia region, a supply route into Crimea, Russian-installed authorities summoned men of fighting age to mobilise, Ukraine's military general staff said in a statement.
Russia has launched dozens of attacks from multiple rocket launchers since Wednesday, the general staff said, along with 16 air strikes and seven missile attacks.
Reuters was unable to immediately verify battlefield reports from either side.
Hundreds of kilometres away, across Ukraine's northern border, Russian soldiers were preparing for fighting in winter conditions by taking part in tactical training exercises in Moscow's close ally Belarus, the Russian defence ministry said.
A flurry of Russian diplomatic and military activity in Belarus in recent weeks has revived fears that Moscow is pressing its ally to become more involved in the Ukraine war.
President Alexander Lukashenko, who relied on Russian troops to put down a popular revolt two years ago, has allowed his country to serve as a staging ground for Russia's invasion of their common neighbour.
He has so far kept his own army from joining in but recent weeks have seen increasing signs of involvement in Belarus from Moscow.
On Saturday, Russian Defence Minister Sergei Shoigu flew unannounced to the capital Minsk.
He and Belarusian counterpart Viktor Khrenin signed amendments to the two countries' security cooperation agreement, without disclosing the new terms.
Thousands of Russian troops have deployed in Belarus since October, Ukraine says, and Belarus authorities have increasingly spoken of a threat of what they call terrorism from partisans operating from across the border.
