The strikes came in the wake of a Ukrainian attack on a plant producing electronics for missiles in Russia's border Voronezh region on Monday that the local governor said killed five people and injured dozens.
Meanwhile, Ukrainian President Volodymyr Zelenskiy has sought support from Western allies for a peace deal while also pushing for fast-track admission to the European Union.
Two people sought medical help after Russian forces struck the southeastern region of Zaporizhzhia, Governor Ivan Fedorov said on Telegram, and three more people were wounded in Sumy, in the north, late on Monday, emergency services said.
Zelenskiy said three people were killed in Sumy, two in Zaporizhzhia and "tragically, the Odesa region has also seen lives lost".
"A Russian drone killed a child, a grandmother, and a man. The mother and two other children from the same family were wounded. Their home was destroyed," he wrote on X.
One woman was injured in a drone attack on Ukraine's second-largest city of Kharkiv, Mayor Ihor Terekhov said on Telegram on Monday evening.
Early on Tuesday, Kyiv authorities briefly issued an air raid alert before withdrawing it.
Zelenskiy warned last week that Russia was preparing a massive attack - something Moscow has said it would conduct regularly. Russia's full-scale invasion of Ukraine started in February 2022.
Meanwhile, Ukrainian attacks on maritime logistics and supply roads have sparked a fuel crisis in Russia and areas of Ukraine it controls.
Kyiv has also intensified air strikes on Russian energy infrastructure, hitting targets as far away as Siberia, more than 2000km from the front line, and undermining the availability of gasoline and diesel in Russia, the world's third-largest oil producer.
The fuel crisis has spread from Russia-annexed Crimea to regions in the east and has also covered Omsk in southwestern Siberia close to Kazakhstan's border.
The Omsk region about 2500km southeast of Moscow is limiting fuel sales, its governor said on Telegram.
The move was "to avoid artificially creating panic buying at gas stations and speculation," Vitaly Khotsenko said in a post on the platform on Monday evening, adding that sales of gasoline would be limited to 40 litres per car and diesel to between 80 and 200 litres, depending on location.
Sales for use in refuelling cans would be banned, he said.
Russia's war on Ukraine has prompted Europe to increase defence spending and partner with Kyiv on possible drone production.
The conflict also spurred Sweden and Finland, until recently members of the EU but not NATO, to join the alliance.
Foreign fighter jets escorted Russian strategic missile-carrying bombers during their 16-hour flight - which included air-to-air refuelling - in the neutral zone over the Barents Sea and the Norwegian Sea, Russia's defence ministry said on Tuesday.
Russia borders NATO members Norway and Finland. The defence ministry did not provide details on the origin of the foreign jets.
with DPA
