Is it transient synovitis a common cause of knee and foot pain one week after a cold in children?

Synovitis. yes, it is but you should have your child examined to be sure it is in fact synovitis and not a more serious infection.
Knee pain. Children often have leg pains and foot pains after or during a viral infection. If there is no swelling or limp, it is not likely to be transient synovitis, which is fairly dramatic.