The question says it all: is there a way to go about removing a scratch from a mirror on a car? We're not talking about an enormous scratch; just big enough to be annoying. Is there a way to sand down the glass next to the scratch to remove all traces of it? Or a glass filler that will make it less evident? Or is this a job that requires replacement?
To remove scratches from a mirror, apply white toothpaste or polishing powder to a damp towel or rag and rub gently at the scratch until it is completely gone. If the scratches are too deep or are on the reverse of the mirror in the silvered surface, then a professional may need to handle the job.
Glass filler won't make it less noticeable, while sanding needs to be followed by polishing.
It's less trouble and less expense (still need materials for sanding and polishing) to replace it with a junkyard part.
Removing the scratch via filling or polishing is sub optimal. Filling will not really fully resolve the issue and polishing will be labor intensive enough that either you will be unhappy doing it or will pay a bit. If you have a remotely mass produced vehicle it would be recommended to checkout junk yards or seek a new replacement.