You could probably save yourself a lot of work if you add more white sheets to the background to cover pretty much everything that can be possibly covered, that might be in your shot.
Coming specifically to your question, you'll have to mask out the background and reveal a solid white layer underneath (or something that matches your video background, if you choose not to take it to a clean white). For the sides, it's probably best to keyframe a mask manually.
For the arm sticking out beyond the white background, you could try using the Roto Brush tool to automate the masking, failing which you'll have to manually keyframe a mask for that too.
If it were me, I'd use that red arm on that light, track it, mask out a copy of the background and slide it behind that light arm.
Looks fairly straight forward, though I can't tell how much slide you have.
Would you by chance know of a video showing this?
It doesn't slide too much, more or less what you see above both left and right.
Thank you for responding!
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There are numerous videos on this, here is one that gets to the gist. It says face tracker but basically you'd mask that light arm in same fashion.
Once you track it, that will allow you to place a background beneath the light that matches your white. I'd just chop a piece of that white part out, give it a good feather on left side and where it meets desk.
Parent the inserted background to the tracked mask for the light arm. That should about do it.
Keep in mind, there are numerous ways to do this.
You could also just use the AE Camera tracker which is pretty solid. In either case, you will want that light arm and the desk masked out on a top layer in order to place the white piece beneath.
The video you linked doesn't work but I think I get the gist of what you're describing,
if you wouldn't mind re-posting the video so I have something to base your comments on I'd appreciate that!