When a box is moving it experiences a frictional force of 20n to the left, or toward the mover. what force does the mover need to apply in order to keep the box moving to the right at 1 m/s?

If there's a friction force of 20N toward the left, then you need to apply
a force of 20N toward the right, in order to keep the box moving at a
constant speed.

When there's a force pointing to the left and an equal force pointing
to the right, then the forces on the box are "balanced" ... they add up
to zero, and the box has zero acceleration.  That means it moves in a
straight line at a constant speed.