Answer:
When the child applies force in the wagon, yes, the wagon also applies force in the hand of the child, but the wagon accelerates because it needs a lot of les force in order to accelerate, and the force that the wagon does in the hand of the child is not enough to make ani effect on the hand.
A similar example of this is: if you step in something with low friction near a wall, and you start to push the wall, you may see that the one moving now is you. This is because the force that you are applying in the wall is not enough to move the wall, but the force that the wall is returning to you is enough to move you.