The boy's name would not be released until the boy's extended family had been notified, the spokeswoman said.
Eastern district police Senior Sergeant Greg Brown said the police serious crash unit was investigating.
"We have been advised by KFC that there's nothing on their cameras, although we will review those as part of our investigation.
"[It's] just an absolute tragedy for the family and all concerned."
St John Hawke's Bay district operations manager Stephen Smith said they were called to the medical centre around 2.40pm.
"We had a call from the medical centre in Napier, the child [which] had the accident had been picked up and taken there.
"We had an ambulance there within three minutes. They called for additional resources and between ambulance staff, paramedics and the medical centre staff, they continued to try and resuscitate the child for approximately half an hour but unfortunately the child died at the medical centre."
Mr Smith said he could not provide any detail on the injuries suffered by the child.
"It is a police issue and also an issue now that is before the coroner."
The car which hit the child was not being driven by a family member, he said.
A woman who answered the phone at the KFC outlet said she "can't answer any questions", before hanging up.
Police appealed for witnesses to the tragedy to contact them.
