Although the operation was a success, the 24-year-old will not know when he will be able to return to the pitch until a check-up in early June.
The former Swansea City forward was taken ill during training on Friday, costing him the chance of facing off against his old side, with whom City drew 0-0 on Saturday.
Following their 1-0 win over West Brom, Mancini told reporters: "We are very sorry for Scott but we hope he recovers quickly.
"He is in hospital, he is getting better and it is under control."
Sinclair has made just two Premier League starts since joining last season's championsand reports have linked him to a return to the Liberty Stadium after Mancini admitted that it would be better for him to leave.