Outlining the aims of the incursion for the first time, President Volodymyr Zelenskyy says Ukraine needs a border “buffer zone.”
Ukraine says it has destroyed a second bridge over the Seym River, which meanders through Russia's Kursk region.
It is the second attack on a bridge in the region in a matter of days and comes as Ukraine presses forward with a cross-border offensive that began on August 6.
On Friday, he said he had crashed into a bridge in the Russian town of Glushkovo.
“Let’s just cut one more bridge,” Ukrainian Air Force Commander Mykola Oleshchuk wrote on Telegram on Sunday, posting an aerial video of an explosion ripping through the bridge near the Russian town of Zvannoye.
He said the attacks were designed to disrupt Russian supply lines.
“The Air Force aviation continues to deprive the enemy of logistical capabilities with precision air strikes,” Oleshchuk said, without giving a date for the attack.
Hours after the attack, Ukrainian President Volodymyr Zelenskyy praised the military incursion and stated its objectives for the first time.
“Our guys are doing a great job in all areas,” he said in his Sunday evening address.
“Our main task in defensive operations is to destroy the maximum Russian military potential and carry out the maximum number of counter-offensive actions. This includes the creation of a buffer zone on the territory of the aggressor – our operation in the Kursk region.”
Pro-Kremlin military bloggers acknowledged the destruction of the first bridge near Glushkovo, which is about 12 kilometers (7.5 miles) north of the border with Ukraine, saying it would disrupt supply lines.
Zvannoe lies 8 kilometres (5 mi) further northwest.
According to Russian news site Mash, the strikes left only one bridge in the area intact, which could further complicate Moscow's attempts to replenish its forces and evacuate civilians.
kyiv has so far said little about the targets of the surprise raid, which began with tanks and other armored vehicles, the biggest attack on Russia since World War II.
Ukraine's military chief, Oleksandr Syrskii, said last week that his forces had advanced across 1,000 square kilometers (390 square miles) of Kursk, although the extent of their control could not be independently verified.
The raid has helped boost Ukrainian morale as Russia advances in the eastern Donetsk region.
On Sunday, Moscow claimed to have captured the village of Svyrydonivka, about 15 kilometers from Pokrovsk, which is a key logistical hub for Ukrainian troops and cities on the eastern front.
Zelenskyy has urged Kiev's allies to lift remaining restrictions on the use of Western weapons on targets deeper inside Russia, including Kursk, saying their troops could deprive Moscow “of any ability to advance and cause destruction” if granted sufficient long-range capabilities.
Ukraine says bases inside Russia have been used to launch long-range attacks on Ukrainian cities and energy infrastructure, causing significant damage and casualties.
On Sunday, Moscow attacked the capital kyiv with ballistic missiles for the third time this month, according to the head of the city's military administration, Serhiy Popko.
Zelensky said Russia had launched more than 40 missiles, 750 guided aerial bombs and 200 attack drones at Ukrainian towns and cities in the past week alone.
