Wheater picked up his fifth booking of the season in Saturday's 1-0 defeat at Coventry, and will consequently have to sit out the game with the Bluebirds in four days time.
With Stephen McManus nursing a long-term ankle injury, Mowbray is short of options as he attempts to find a new defensive partner for Matthew Bates.
However, in fit-again Seb Hines and full-back Justin Hoyte, the Boro boss feels he has two alternatives who can fill the void.
"David is suspended and there's very little we can do about it," said Mowbray. "I know we have Stephen and David out, but you just have to cope.
"We will work on different things this week if the weather is okay and see who we can put in there with Matthew.
"I have Seb Hines, but I also have Justin Hoyte who I could use, with his high fitness levels and energy. We have options, and I am not going to cry over it.
"We want to be competitive next week. It will not be easy against (Craig) Bellamy and (Jay) Bothroyd, but that's what it is and we have to compete."
Barry Robson should be available to face Cardiff after missing last weekend's game through illness, and Kevin Thomson could also return after a prolonged break in an attempt to protect his leg.