The cart will move with a constant velocity if no net external force acts on it. When the horse applies a force on the cart, frictional force also acts between the tyres of the cart and road to oppose the motion. The cart will move with a constant velocity only if the force applied by the horse is equal to the frictional force.
Moving at constant speed is called stopping an object or moving at the same speed. Yeah, she's got to move, that horse. If it accelerates or slows down, it does not move at constant speed.
