2 Answers
If you shoot an evenly lit white frame you could invert it, adjust the levels so that the non-affected areas of the frame were completely black, and then add the result to the original footage. It might take some tweaking of the levels to get it right.
It depends, which software you want to use. The easiest one is probably After Effects, which has a function to replace an object completey by analyzing the background.
But you could as well try to set a new alpha layer on the film over its complete length, masking the dark spot and light the spot area up a tiny little bit.
This might result in a loss of contrast and sharpness, though.