In a surface, the force to make an object just start is 4N. A force of 4.1N is applied on an object. Will the object move a bit and stop or will it...

The force required to just make the stationary object
start moving is 4N. This is due to the force of static friction between the object and
the surface it is placed on. To move the object, this force of friction has to be
overcome.

Once the object has been set into motion, there
is a force of kinetic friction between the object and the surface. In most cases the
coefficient of kinetic friction is less than that of static friction, but there is a
force opposing the motion nevertheless.

From the way you
have framed your question I think you are applying the force instantaneously and not
over a prolonged duration of time. The force applied will make the body start moving but
it will move only for a small duration of time which is required for the force of
friction to bring it to a halt.