Thanks for your comments, gentlemen. I actually cleaned the image sensor with the feature in the menu, and my oh-so troublesome dots vanished! I'm really happy it's only a small issue. Apparently from time to time the pixels require re-aligning; I wasn't aware of this. Just one of those things you find out as you progress, I guess.
Sorry, maybe I could have explained better. What I mean, is that the cut out seems to float around in the video, because it isn't tracked to the new video. It's basically sitting on top, and doesn't appear to be part of the video because of it. I can track it frame by frame, but even then it doesn't look right; maybe After Effects is the only way.
I'm afraid I don't currently have AE, but thank you for your reccomendation, anyway. :)
I did figure After Effects would be preferable, considering Premiere is (I think) barely considered compositing software.
Would it be impossible to use Premiere, or would it just be quite difficult? I don't have AE, since it's quite pricey...:(