I'm new to Inkscape and it has been working well for me so far. However, I am trying to move a rectangle portion of a completed image a little bit to the left or right to achieve a "glitched" look. Is there any possible way to do this? Any help is appreciated.

There's no one step method, but if the target is not too complex you can duplicate all of it, clip it with the rectangle and shift the clipped duplicate. You can simulate transparency underneath by using another clipping path on the original, this time to cut away the 'window' in a frame-shaped path.
Experiment with a simple group to figure out how clipping works, it's quite simple, even in this particular case.