Roto-Scope! That's what you want to do. It's called roto-scoping. Several apps out can do that. Studio Artist, Shake, there's a "cheat" you could do in FCP.
Place a copy of the clip directly above itself. Apply a "Mask Shape" filter (Matte Bin) to single out a small space of the frame, something close to the white spot, but something to cover the white spot. And then a "Mask Feather" (same Bin) to make it soften out to the surrounding video on the track below.
Then, move the "Center" variables in the Motion Tab in the Viewer of that clip with the Mattes. Type in small numbers, what what it does, you'll get it, it's super simple. Now, move that masked out frame on top, so it covers the white spot area of the bottom layer video.
You'd have to maybe do some keyframing, or a lot. And it's a lot to explain. Basically what you'd do in Photoshop, more or less, but with motion keyframes. You'll keyframe the "Center" variable in the Motion Tab in the Viewer.
I don't do tons of roto-scoping myself, I have occasion to farm it, out but you want to look into applications that can do "roto-scoping". I'm drawing a blank at the moment, but there's several out there. What I explained above is a gorilla-video cheat. And can sometimes work ok. But you'd really rather have an application that actually does "roto-scoping". Go west young man, or, um, south, I dunno, whatever... roto-scope...