How much energy? The wormhole doesn't have an inherent mass, so how much energy does it take? After all you can put an arbitrary amount of mass through it.
You're looking at it the wrong way. If you can extract energy from dropping the rock through the wormhole, it takes energy to pull the rock away from the wormhole (and, by symmetry, it takes that much energy to move the wormhole away from the rock).
I would guess that that would depend on the situation.
I would think that whatever method you have of moving a mouth of the wormhole, would have some interactions with the masses providing the gravitation.
Also, I'm not 100% sure that a wormhole would not have something like an inherent mass, because iirc there are solutions where a black hole mass is entirely because of the way spacetime is set up, without any particles needed? I'm not sure about that, but I think that is what I read..
