AMNA NAWAZ: Today, Ukrainian troops fell back to defensive lines amid one of Russia's largest offensives since the beginning of the full-scale invasion more than two years ago.

President Volodymyr Zelenskyy has postponed all foreign travel, and the U.S. secretary of state announced new weapons deliveries while in Ukraine.

But, as Nick Schifrin reports, in Northeast Ukraine's Kharkiv, the front is fragile.

NICK SCHIFRIN: Along the Ukraine-Russia border, Ukrainian troops try to hold the line.

This is the Kharkiv region, where Ukraine long ago pushed Russia out.

Now it needs reinforcements to hold Russia back.

Vovchansk is just two-and-a-half miles from Russia, and the center has been ripped apart.

A Ukrainian soldier fighting in the city sent us videos of Russian shelling and Russian troops on the city's edge.

He says, with the help of drone attacks on Russian tanks today, they push Russian troops back.

But he still fears Russia could take the city.

TAMAZ GAMBARASHVILI, Vovchansk Military Administration, Ukraine (through translator): The situation in Vovchansk is very dire.

For the sixth day in a row, the enemy has been bombarding the city with all possible weapons and munitions, rockets, artillery and bombs.

NICK SCHIFRIN: Tamaz Gambarashvili is the regional administrator in Vovchansk.

He says the defensive lines built in other areas, including trenches and anti-tank dragon's teeth aren't as robust near Vovchansk.

TAMAZ GAMBARASHVILI (through translator): After the deoccupation in September of 2022, military and engineers built defensive lines.

But because we are so close to the Russian border, and because of the constant fighting and constant shelling, perhaps we were not as successful in putting these defensive lines in as we were in other areas.

NICK SCHIFRIN: The Russian offensive that began late last Thursday has captured more territory than any point since February 2022.

U.S. and Ukrainian officials say Russia's goals are to push far enough into Kharkiv region to be able to reach Kharkiv City with artillery and to draw Ukrainian forces away from fierce fighting in the Donbass.

Inside Vovchansk, soldiers go door to door to try and convince villagers to evacuate, including Natalia Yurchenko, who's been in hiding.

NATALIA YURCHENKO, Vovchansk Evacuee (through translator): For five days, we never left the house.

We did not see anyone.

We were so afraid to go out, we never even opened the door.

ANDRII SEMENKO, International Rescue Committee: It's pretty hectic, and there are a lot of shellings, and it's almost everywhere in Vovchansk.

NICK SCHIFRIN: Andrii Semenko is a humanitarian worker with the International Rescue Committee.

The IRC and other aid groups provide medical and psychological support to thousands of the newly displaced, including the most vulnerable.

ANDRII SEMENKO: Some of the children, their moods are up and down, and they might start crying straight away without any reasons.

From the sound of if the door closed kind of very loudly, they would get like a fright reaction, and they would -- some of them burst into tears, and you would have to calm them down, so that you are safe and you're OK. NICK SCHIFRIN: Kharkiv City's scars from Russia's 2022 campaign haven't yet healed.

But, today, the buildings bear new wounds and residents flee with only their most precious belongings.

MAKAR, Kharkiv Resident (through translator): There was an explosion, then a cloud of dust.

This is what our life looks like.

Our building was hit, but we will get over it.

NICK SCHIFRIN: In Kyiv today, Foreign Minister Dmytro Kuleba and Secretary of State Antony Blinken paid tribute to Ukrainians killed fighting Russia since its initial invasion in 2014.

ANTONY BLINKEN, U.S. Secretary of State: I'm here as part of a show of support.

NICK SCHIFRIN: And Blinken announced $2 billion of additional weapons deliveries.

Ukrainian officials watched Russia's buildup on the Kharkiv border, frustrated by U.S. demands not to fire U.S. weapons into Russia.

Today, Blinken said that policy wouldn't change.

ANTONY BLINKEN: We have not encouraged or enabled strikes outside of Ukraine, but, ultimately, Ukraine has to make decisions for itself about how it's going to conduct this war.

NICK SCHIFRIN: U.S. officials have faith that Ukraine can protect Kharkiv City.

But weapons and manpower shortages means it remains vulnerable, and Russia is determined to keep inflicting pain.

For the "PBS NewsHour," I'm Nick Schifrin.