A Russian-installed official said Ukraine had recaptured Piatykhatky, a village in the southern Zaporizhzhia region, and were entrenching themselves there while coming under fire from Russian artillery.
"The enemy's 'wave-like' offensives yielded results, despite enormous losses," the official, Vladimir Rogov, said on the Telegram messaging app.
Russia's defence ministry made no mention of Piatykhatky in its daily update, in which it said its forces had repelled Ukrainian attacks in three sections of the 1000 km front line. A separate statement from Russia's Vostok group of forces said Ukraine had failed to take the settlement.
Reuters could not independently verify the battlefield reports.
There was no comment from Ukraine, which last week said it had recaptured another nearby settlement, Lobkove, and a string of villages further east, in Donetsk region, at the start of its long-awaited counteroffensive.
Ukrainian officials have imposed an information blackout to help operational security, but say that Russia has suffered much greater losses than Ukraine has during its new assault.
A regional official said Ukrainian forces had destroyed a major Russian ammunition dump in occupied Kherson region, part of a weeks-long effort by Kyiv to wreak havoc with Russian supply lines.
British defence intelligence said heavy fighting in recent days had been focused on Zaporizhzhia, western Donetsk and around Bakhmut, which Russian mercenaries captured last month after the longest battle of the war.
"In all these areas, Ukraine continues to pursue offensive operations and has made small advances," it said on Twitter.
Russian defence operations had been "relatively effective in the south", with both sides suffering heavy casualties, the assessment said.
Russian President Vladimir Putin, who rarely comments on the course of the war, made two unusually detailed interventions last week in which he derided the Ukrainian push and said Kyiv's forces had "no chance" despite being newly equipped with Western tanks.
His comments appeared intended to reassure Russians at a crucial juncture, nearly 16 months into the conflict, as Ukraine seeks to break months of virtual stalemate and take back the 18 per cent of its territory that remains under Russian control.
At talks in St Petersburg on Saturday, South African President Cyril Ramaphosa presented Putin with a 10-point peace initiative from seven African countries and told him the time had come for Russia and Ukraine to start negotiations to end the war.
Putin responded by rattling off a string of familiar accusations denied by Ukraine and the West and saying it was Kyiv, not Moscow, that was refusing to talk. He thanked Ramaphosa for his "noble mission".
Russian news agencies quoted Kremlin spokesman Dmitry Peskov as saying Putin had shown interest in the plan but it would be "difficult to realise".
In Kyiv the previous day, Ukrainian President Volodymyr Zelenskiy had told the African delegation - the first since the start of the war to hold separate face-to-face talks with both leaders on their peace initiative - that allowing negotiations now would just "freeze the war" and the suffering of the Ukrainian people.
The vast gulf between the two sides was further underlined when Putin used a flagship economic forum on Friday to slur Zelenskiy personally and to restate the objectives of "demilitarising" and "denazifying" Ukraine that he set out on day one of the war, and which Kyiv and the West reject as a false pretext for invasion.
However, Ramaphosa sought to cast the trip to Ukraine and Russia in a positive light, tweeting on Sunday that the "Africa Peace Initiative has been impactful and its ultimate success will be measured on the objective, which is stopping the war".
