Central defender Clarke was forced off after an early challenge with Birmingham striker Jesse Lingard at the John Smith's Stadium and has damaged ligaments.
"The incident set the tone," said disappointed manager Mark Robins after seeing his side slip to a fourth defeat in five league matches. "The pass was horrible to him and he came off second best, which he doesn't do very often.
"He has injured his medial ligament and is looking at between four to eight weeks out."
Striker Martin Paterson and midfielder Jonathan Hogg both missed the game due to injury.
Robins added: "Martin has a hand injury and will be fit next week, however that falls when he can play for Northern Ireland and not us. He hasn't trained in the last week.
"Hoggy tried his knee (on Friday) and I expected him to be fit, but when he tested it he still felt pain.
"We missed him and his energy, but that happens and it is about how you cope with it and we didn't."