The Northern Ireland international is on crutches having broken a bone in his foot in the closing stages of Saturday's 1-1 draw with Accrington, ruling him out for the remainder of the season.
York manager Nigel Worthington told Press Association Sport: "He's broken his fifth metatarsal bone in his foot and is going to be out of action for the next six to seven weeks.
"It's disappointing, but we have to live with it. I've been in the game long enough to accept that these things happen.
"I'm very disappointed for the boy and ourselves. He's been doing very well for us, but the door opens for someone else now."
Midfielder Carson, 19, made five appearances for the Minstermen, starting every game for Worthington's side since arriving from Town last month.