Must we wash what is touched by a child who does not avoid impurity, or is he considered among those who frequent us, like a cat?

The basic principle is that a child's hand is pure, and anything it touches does not need to be washed unless we are certain of its impurity. However, if the hand was wet when it touched something, or if what it touched was wet, then what it touched must be washed. Prayer is not valid before removing such impurity.