Rescuers were working to restore power in the town with a pre-war population of about 83,000. Regional officials said there had been 10 explosions during the attack and Mayor Artem Semenikhin said the power system was in critical condition.
"At the moment, energy workers are doing everything they can to provide electricity to the hospital and the water supply system," he said in the early hours of Thursday.
Hospitals continued to operate, he added.
Konotop is in the Sumy region, which Russia has particularly targeted in recent weeks.
Ukraine used Sumy as a staging ground for a surprise incursion into Russia across the border last month in a bid to try to seize the battlefield initiative as Russian forces inch forward in the east of Ukraine.
After making rapid progress initially, Kyiv's advances into Russia's Kursk region appear to have stalled, and a senior Russian commander said on Wednesday his forces had taken back control of about 10 settlements in a counterattack.
Ukraine has not commented on the latest Russian-reported gains. Reuters was not able to independently verify battlefield developments.
The strike on Konotop was part of a broader Russian attack using 64 drones, the air force said. It shot down 44 of them over nine different regions, it said.
Such attacks have become regular, almost nightly occurrences since Russia invaded Ukraine in February 2022.
Though Ukraine is heavily outgunned and lacks a large arsenal of long-range missiles, it has tried to take the fight back to Russia by launching hundreds of drones at it.
Semenikhin said the authorities in Konotop did not know when the power would be restored and they planned to supply water on an hourly basis.
The attack also damaged seven apartment buildings, medical and educational institutions, a shop, a bank and a section of the town's tramway, regional prosecutors and local officials said.
The Russian drone attack also caused a fire at a furniture storage warehouse in the central Khmelnytskyi region which emergency services were still trying to extinguish in the morning, governor Serhiy Tiurin said.
The governor of the northern Chernihiv region said Russia had stepped up its attacks there in the past day, causing damage to the region's infrastructure. He did not say what exactly had been hit.
