Russian troops are focusing their attacks on the city of Lysychansk, after seizing its sister city, Severodonetsk, across the Siverskyi Donets river.
Russian forces were using artillery to block off access to the city from the south, according to the Ukrainian General Staff. Civilian and military infrastructure had also been hit, the report said, although this could not be verified independently.
Within the Luhansk region, Ukrainian troops now only control Lysychansk, although Russian soldiers have already advanced to the city's outskirts.
Earlier, a nine-story apartment building in Kyiv was struck in a missile attack, injuring at least four people and leaving others trapped under the rubble, Kyiv Mayor Vitali Klitschko said.
While fighting is raging in eastern Ukraine, Kyiv had been largely has been spared from the violence in recent months. It last faced a barrage of missiles three weeks ago.
Anton Herashchenko, an advisor to the Ukrainian interior minister, wrote on Telegram that another rocket hit the grounds of a kindergarten in the capital's Shevchenko district.
A 7-year-old girl and her mother were found alive in the debris, Klitschko said. The building is located in the immediate vicinity of the Artem weapons factory, which has been attacked several times previously by Russia.
Ukrainian member of parliament Olexiy Honcharenko said a total of 14 rockets were fired by Russian troops across the city and surrounding area early Sunday.
Moscow's forces also targeted the central Ukrainian region of Cherkasy, killing one person and injuring five, Ukrainian officials said. Two missiles came down near the regional capital on Sunday, according to regional governor Ihor Taburets.
A video shared on social media showed a railway bridge across the Dnipro river apparently damaged by the strikes. Ukraine's railway operator announced it was temporarily halting regional train services around Cherkasy.
Meanwhile, pro-Russian separatists said they have evacuated 250 more people from the air-raid shelters of the Azot chemical factory in Severodonetsk after Ukraine gave up the city.
Some 200 civilians, including young children, had been brought out of the plant on Saturday, the Luhansk separatist representative in Moscow, Rodion Miroshnik said on Telegram.
However, it was unclear where they had been taken. There had also been mention of a far greater number of evacuees but no clearer information was available.
The Ukrainian military governor of Luhansk had said there were 568 civilians sheltering in the bunkers of the plant, before troops withdrew.
The latest attacks came as G7 leaders convened in southern Germany to discuss increasing pressure on Russia to end the war.
German Chancellor Olaf Scholz has condemned Russia's "brutal" missile attack on Kiev, saying that it reinforced the need for G7 leaders to "stand together and support the Ukrainians in defending their country."
Proposals to jointly increase pressure on Russia, collect military and humanitarian support for Ukraine, and the impact of the conflict in driving energy and food prices sharply higher will be focal points at the summit.
G7 leaders have already used the summit as a forum for the announcement of a ban on imports of Russian gold, while talks about a possible oil price cap are also underway.
Ukrainian President Ukrainian President Volodymyr Zelensky is due to address delegates via video link during the talks.
It also emerged that Russian Defence Minister Sergei Shoigu has visited Russian soldiers fighting in Ukraine, although it was unclear exactly where he went.
Russian President Vladimir Putin is also planning travels in the coming days, reportedly to Tajikistan and Turkmenistan in what would be his first trip since the beginning of the war.
Putin will first fly to Tajikistan, a military ally, then to Turkmenistan to join a summit of the Caspian Sea states planned for Wednesday, an official told TASS news agency.
