Question

Can an object have mechanical energy without having momentum?

Answer 1

Quick Review:
-Mechanical energy is the net sum of potential energy and kinetic energy.
-Momentum, on the surface, is just mass x velocity or P=MV

Therefore; an object with no motion (v=0) will also have a momentum of zero. Furthermore, kinetic energy will also be zero without motion.

However, potential energy does not care about motion so having zero momentum is not enough to automatically assume mechanical energy is also zero. So the answer is yes.