Russia carried out its third large-scale bombardment of Ukraine in a week on Monday, with explosions ringing out early in the morning in Kyiv and several other cities after a volley of missiles was fired on the first day of the school year.

The attacks coincided with intense fighting along the frontline in the eastern part of Ukraine, and it came a day after Ukraine launched drones at refineries and electrical plants near Moscow and elsewhere in Russia, causing fires and other damage.

The Monday attack came around 5:30 a.m., and an air raid alert was lifted an hour later. At least two people were injured.

Schools in Ukraine must provide bomb shelters for students or operate only online, under a law that came into force after Russia invaded Ukraine in 2022.

In cities near the front, nearly all education takes place online, but many schools are open in Kyiv, and children attend even on days when missiles have targeted the city.

Serhiy Popko, the military administrator in Kyiv, said in a post on social media that Russia had fired about 10 ballistic missiles at the city and more than 10 cruise missiles.

The cruise missiles, which are slower than ballistic missiles but can change direction during flight, maneuvered for about two hours before setting a course toward Kyiv, arriving simultaneously with the ballistic missiles.

The attack stood out for the large number of ballistic missiles fired at Kyiv, Mr. Popko wrote, but early reports of damage were minor.

Debris from intercepts by antiaircraft missiles over the city fell in four districts, setting four cars on fire and damaging commercial buildings, Kyiv’s mayor, Vitali Klitschko, said in a statement. Mr. Klitschko wrote that two people had been wounded.

The authorities also reported explosions in Kharkiv, not far from the Russian border in Ukraine’s northeast.

Before the attack on Monday, President Volodymyr Zelensky of Ukraine said that Russia had fired more than 160 missiles of various types, 780 guided aerial bombs and 400 attack drones at Ukrainian targets over the preceding week.

Ukrainian officials have appealed to allies to allow the use of long-range weapons provided by the West, such as rockets and cruise missiles, to strike back at targets in Russia.

Ukraine’s military is stepping up strikes with domestically produced drones. On Sunday, Russia’s military claimed to have thwarted a major Ukrainian drone attack, but fires at refineries and power plants suggested that at least some had gotten through.