Ukrainian President Volodymyr Zelensky has warned Moscow it is sowing a deep hatred for Russia among his people, as constant artillery barrages and bombings reduce cities to rubble, kill civilians and leave survivors to scrounge for food and water to survive.

“You are doing everything so that our people themselves leave the Russian language, because the Russian language will now be associated only with you, with your explosions and murders, your crimes,” he said in an impassioned video address.

Russia’s invasion of Ukraine has ground into a war of attrition in many places, with the toll on civilians rising as Moscow seeks to pound cities into submission from entrenched positions.

A nuclear research facility in the besieged city of Kharkiv, near the Russian border, again came under fire on Saturday, and Ukraine’s nuclear watchdog said it had been impossible to assess the extent of the damage.

Kharkiv has been besieged by Russian forces since the start of the invasion and has come under repeated shelling that has hit residential buildings and critical infrastructure.

Ukrainian authorities previously said Russian shelling had damaged buildings at the facility, but there had been no release of radiation.

The International Atomic Energy Agency has said the nuclear material in the facility is always subcritical and the inventory of radioactive material is very low, reducing the risks of radiation release.

On the western side of the country, Russian rockets struck Lviv on Saturday while US President Joe Biden visited neighbouring Poland, serving as a reminder that Moscow is willing to strike anywhere in Ukraine despite its claim to be focusing its offensive on the country’s east.

Early on Sunday, a chemical smell still lingered in the air as firefighters in Lviv sprayed water on a burned section of an oil facility hit in the Russian attack.

Russia’s air strikes shook the city that has become a haven for an estimated 200,000 people who have had to flee their home towns. Lviv had been largely spared since the invasion began, although missiles struck an aircraft repair facility near the main airport a week ago.

Two cities at opposite ends of the country are seeing some of the worst suffering: Chernihiv in the north is strategically located on the road from the Belarusian border to the capital Kyiv, and Mariupol in the south is a key port city on the Sea of Azov.

Both are encircled by Russian forces, but still holding out.

Chernihiv has been under attack since the early days of the invasion and over the last week, Russia destroyed the main vehicle bridge leading out of the city and rendered a nearby pedestrian bridge impassable, cutting off the last route for civilians to flee, or for food and medicine to be brought in.

More than half of the city’s 280,000 inhabitants have already fled and hundreds who stayed have been killed, mayor Vladyslav Atroshenko said.

Russian forces have bombed residential areas from low altitude in “absolutely clear weather” and “are deliberately destroying civilian infrastructure: schools, kindergartens, churches, residential buildings and even the local football stadium”, he told Ukrainian television.

Refugees from Chernihiv who reached Poland this week spoke of broad and terrible destruction, with bombs flattening at least two schools in the city centre and strikes also hitting the stadium, museums and many homes.

They said that with utilities knocked out, people are taking water from the Desna river to drink and that strikes are killing people while they wait in line for food.

Previous bombings of hospitals and other non-military sites – including a theatre in Mariupol where Ukrainian authorities said a Russian air strike is believed to have killed 300 people last week – have prompted allegations of war crimes.

The invasion has driven more than 10 million people from their homes, almost a quarter of Ukraine’s population. More than 3.7 million have fled the country entirely, according to the United Nations. Thousands of civilians are believed to have died.