The centre half required a minor operation on Sunday night to repair a deep cut he received on his wrist when tumbling into a sponsor board in the second half of City's match at Gillingham on Saturday.
Parent club West Ham United arranged for a plastic surgeon to look at Burke's injury after the youngster had received initial treatment pitch-side from City's physios and then at a hospital near to Gillingham's Priestfield Stadium on Saturday evening.
Burke, who has now returned to West Ham following the completion of his latest youth loan deal, has been ordered to rest for two weeks to allow the wound to heal properly.
Even upon his return to action, Burke is expected to have to wear a cast on his wrist for his first few games back.