Question

A horse continues to apply a force in order to move cut with a constant speed explain why

Answer 1

Yes its true that in idle state anything must remains in its original condition if it is in motion always remains in motion until and unless an external force is applied.
But here in this case there is a resisting force (frictional force) which tends to slow down the speed of the cart thus to overcome this resisting an equal and constant force is require to make the cart run with constant speed