The 19-year-old defender suffered the problem during Saturday's 3-1 defeat to Ipswich Town at Portman Road.
"Unfortunately scans show that Mitch has broken two bones in his hand," Birmingham boss Lee Clark is quoted as saying by Sky Sports News.
"We won't be risking him for our last league game, although we would have done so if it had been a playoff fixture.
"The injury is not too serious and he will be ready to return for the start of pre-season."
Tenth-placed Birmingham concluded their Championship campaign at home to Blackburn on Saturday.