Belgorod regional governor Vyacheslav Gladkov said a mother and her four-year-old son were killed while travelling in a car.
The fire at the Tuapse refinery was extinguished, local authorities said. The extent of damage was not clear.
The refinery had undergone several months of repairs, completed only at the end of April, after a previous fire in January. At the time, a Ukrainian source said that blaze too was caused by Ukrainian drones, which have repeatedly struck Russian refineries and caused significant disruption to output.
Tuapse has an annual capacity of 12 million metric tons (240,000 barrels per day). It produces naphtha, fuel oil, vacuum gasoil and high-sulphur diesel, supplying mainly to Turkey, China, Malaysia and Singapore.
Russia's defence ministry said early on Friday that Russian air defences had destroyed 102 Ukrainian aerial drones and six uncrewed naval boats in the Black Sea overnight.
In Russian-annexed Crimea, an electrical substation was damaged in Sevastopol, local governor Mikhail Razvozhaev said on Telegram. He said there would be "isolated blackouts" and school classes had been cancelled.
Ukraine's military said it shot down all 20 drones launched overnight in Russia's latest such attack on Ukraine since its February 2022 invasion.
(Reporting by Reuters, Writing by Mark Trevelyan, Editing by Timothy Heritage)

A Ukrainian serviceman from the 108th Brigade of Territorial Defence carries a Ukraine-made multi-purpose drone Leleka-100 on a field near a frontline in the direction of Zaporizhzhia, Ukraine, 15 February 2024, amid the Russian invasion. Russian troops entered Ukrainian territory on 24 February 2022, starting a conflict that has provoked destruction and a humanitarian crisis. EPA-EFE/KATERYNA KLOCHKO