We solve this with something called a force body diagram. The child, the Earth, and the wagon can all be diagramed independently.

The wagon has two forces acting on it. The force of the child pulling it, and the force of friction from the ground (maybe air resistance too, if you want). If the pulling force exceeds the friction force, the wagon accelerates.

The child has two forces acting on them. The wagon pulls back with the same force they pull on it (equal and opposite force that this question is asking about). Also, the ground pushes them forward with the same force they push on the ground while extending their legs (the normal force... Also a consequence of Newton's 3rd). If the normal force exceeds the reaction force of the wagon, the child accelerates forward (along with the wagon).

The ground has two forces. The friction with the wagon actually acts forwards on the ground since it works backwards on the wagon. And also the force of the child extending their legs backwards also pushes the ground backwards. We already know that the force from the kid's legs exceeds the reaction force of the wagon in their hand. And we also know them at this pulling force exceeds the friction of the wagon to the ground. Therefore, the kid's legs must be pushing harder than the friction and the Earth must accelerate backwards for the kid to be able to accelerate forwards.

The thing is, the Earth is massive. It will not accelerate very much. Negligibly in fact. So basically, we can if ore it and only focus on the child accelerating.

Edit: crude diagram the forces acting on the cart are in green. The forces acting on the child are in blue. The forces acting on the Earth are in red. Every force has an equal and opposite pair. F for friction. P for pulling. R for running all show up twice in opposite directions (and equal magnitude) and their relation tells you which direction each object is accelerating.