Bostan Care Homes Limited, which operates Woodford Care Home, in Holderness Road, pleaded guilty and was fined £2,800 for the offences and ordered to pay costs of £3,500 and a victims' surcharge of £15.
Following an inspection by environmental health officers in March 2012, the extraction ventilation system in the kitchen was found to be insufficient.
The system had not been improved after several follow-up visits.
It was also found the kitchen was not being managed properly and there was not an adequate food safety management system.
Trevor Todd, assistant head of service for public protection at the city council, said: "Environmental health officers work closely with businesses providing advice and education to ensure that the highest standards of food safety are maintained.
"This is particularly important when catering for vulnerable people. When businesses fail to act upon the advice then it leaves the officers no other alternative but to use graduated enforcement action."
Councillor John Hewitt, portfolio holder for neighbourhoods and communities, said: "It is not acceptable for businesses to receive zero out of five as a food hygiene rating. I would encourage all food businesses to follow the advice given by the food safety team to ensure that their customers are protected from the risk of food poisoning."