Russia's military has appeared to acknowledge that Ukrainian units had driven deep into the country, saying it had hit troops and equipment around 30km from the border.
Kyiv launched a surprise border incursion last Tuesday, shocking Russia, which has sent in additional reserves, tanks, drones, artillery and aviation to try to quash the offensive.
It comes as Moscow and Kyiv also accused each other of starting a fire on the grounds of the Russian-occupied Zaporizhzhia nuclear power plant
The defence ministry in Moscow said that its troops had "foiled attempts by enemy mobile groups with armoured vehicles to break through deep into Russian territory".
But in an apparent sign of how far some Ukrainian units have managed to advance, it said it had hit enemy units near the villages of Tolpino and Obshchy Kolodez, which are around 25km and 30km from the Russia-Ukraine border.
A senior Ukrainian official told AFP that "thousands" of its troops had been deployed to the operation.
Dozens of armoured Ukrainian vehicles daubed with a white triangle have been traversing the border region of Sumy, where Ukraine's army has mounted an incursion inside Russian territory.
Reporters in the border territory witnessed military vehicles of varying types marked with the insignia, apparently used to identify hardware being used by Ukraine for its offensive in Russia's western Kursk region.
Russia's military said it had continued to use aviation, drones and artillery to try to push back the attack.
It also said it had foiled an attempt by Ukraine to break into another border district of the Kursk region, further to the south from where most of the fighting has occurred so far.
Ukrainian President Volodymyr Zelensky said Ukraine launched an incursion into Russian territory to "restore justice" and pressure Moscow's forces, in his first acknowledgement of Kyiv's surprise offensive into the western Kursk region.
Moscow's forces are in their sixth day of intense battle against Kyiv's largest incursion into Russian territory since the start of the war, which left southwestern parts of Russia vulnerable before reinforcements began arriving.
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Russian authorities rushed to evacuate residents and imposed a sweeping security regime in three border regions yesterday, while Belarus, a staunch ally of Moscow, sent more troops to its border with Ukraine, accusing Kyiv of violating its air space.
In his nightly video address, Mr Zelensky said he had discussed the operation with top Ukrainian commander Oleksandr Styrsky, vowing to respond in kind after Russia launched its full-scale invasion in February 2022.
"Ukraine is proving that it can indeed restore justice and is ensuring the exact kind of pressure that is needed - pressure on the aggressor," he said.
Russia's defence ministry said that it had destroyed 14 Ukrainian drones and four Tochka-U tactical ballistic missiles overnight in the Kursk region, and 18 drones over other Russian regions that Ukraine frequently attacks.
In a statement, it called the ground incursion, which military analysts say caught the Kremlin off guard, "barbaric" and said it made no military sense.
Ukraine has at most occupied several tens of square kilometres of Russian territory without laying claim to it, while Russia controls more than 100,000sq/km of Ukraine's internationally recognised territory.
Russia's top general Valery Gerasimov said on Wednesday the attacks had been halted, but Russia has not pushed the Ukrainian forces back over the border.
Russian military bloggers said the situation had stabilised after Russia's reinforcements, though they said Ukraine was swiftly building up forces.
Injuries and evacuations
Kursk officials said 13 people were injured in the city after debris from a destroyed Ukrainian missile fell onto a nine-storey residential building.
An image posted by Kursk's mayor showed flames rising through an apartment block surrounded by charred debris.
It was not clear whether there was further damage.
Moscow and Kyiv rarely disclose the full extent of damage inflicted by attacks on them unless there are injuries or damage to residential buildings.
Alexei Smirnov, Kursk's acting governor, ordered local authorities to speed up the evacuation of civilians in areas at risk.
Russia's TASS state news agency reported that more than 76,000 people had been evacuated yesterday.
Kyiv and Moscow deny targeting civilians in their attacks in the war, which has killed thousands of people and displaced millions of Ukrainians.
Some Russian military bloggers questioned why Ukraine was able to pierce the Kursk region so easily.
After a father and his four-year-old son were killed in a Russian air attack near Kyiv, Mr Zelensky asked Ukraine's Western partners for "strong decisions" allowing his troops to strike deep inside Russia with Western weapons.
"When Ukraine's long-range capabilities have no limits, this war will definitely have a limit," he wrote on X.
Russian Human Rights Commissioner Tatyana Moskalkova said she had sent an appeal to the United Nations demanding it condemn Ukraine's actions in Kursk.
In a Telegram post, Ms Moskalkova said she was asking the UN Human Rights commissioner to "take measures to prevent gross mass violations of human rights".
Zaporizhzhia fire
Meanwhile Moscow and Kyiv accused each other of starting a fire on the grounds of the Russian-occupied Zaporizhzhia nuclear power plant, as Ukraine urged residents to remain calm and reported no sign of elevated radiation.
The UN's International Atomic Energy Agency nuclear watchdog, which has a presence at the six-reactor facility, said its experts had seen strong, dark smoke coming from the northern area of the plant in southern Ukraine following multiple explosions.
Ukrainian President Volodymyr Zelensky accused Russia of lighting a fire which he said was visible from the Kyiv-held city of Nikopol which looks out onto the Russian-held plant.
Evgeny Balitsky, a Russian-installed official in the occupied south, accused Kyiv's forces of causing the fire by shelling the nearby city of Enerhodar which, like the plant, was captured by Russia soon after its February 2022 invasion.
The IAEA said there had been no reported impact on nuclear safety at the site.
"Team was told by (the nuclear plant) of an alleged drone attack today on one of the cooling towers located at the site," it wrote on X.
Mr Zelensky published grainy video showing black smoke that appeared to be coming out a cooling tower with a blaze burning at its foot.
"Currently, the radiation indicators are normal. But as long as Russian terrorists retain control over the nuclear plant, the situation is not and cannot be normal," he said.
The Russian management of the facility said emergency workers had contained the fire and that there was no threat of it spreading further.
"The fire did not affect the operation of the station," it said.
The six reactors at the plant located close to the frontline of the war in Ukraine are not in operation but the facility relies on external power to keep its nuclear material cool and prevent a catastrophic accident.
Moscow and Kyiv have routinely accused each other of endangering safety around it.
