Some 2,000 people also have been evacuated from the city of Izyum in eastern Ukraine, the head of Kharkiv Oblast Oleh Syneyhubov said in a Facebook post.
He said they had been brought to safer regions in 44 buses. A video showed bottles of drinking water and other food being loaded into trucks.
Evacuations were also focused on the north-eastern city of Sumy and residents had already boarded buses, evacuating more than 12,000 on Thursday according to the emergency service.
However, it again proved impossible to bring people from the city of Mariupol to safety.
"Mariupol remains completely blocked for both the evacuation of people and humanitarian aid," Ukrainian Vice Governor Iryna Vereshchuk said, according to the Ukrinform news agency.
Fighting was taking place directly on the route out of the city and trucks carrying humanitarian cargo returned to Zaporizhzhia, she said, noting it was the fourth day that it had proved impossible to deliver water, medicine and food.
Vereshchuk said an agreement had been reached with Russia to open an escape corridor in the morning but several attempts had to be called off.
Moscow and Kiev each blamed the other for the failure.
