I believe it may be worth a try, if they're strictly minor surface scratches only seen at certain angles in lighting.
The protector's surface is going to serve as the new reflection source, an the adhesive and glass layers will mitigate the impact of any existing scratches.
Playing around with a busted protector, the adhesive seems like it could have some healing-like properties and "fill-in" the scratches. These things are far from optical grade anyway, and do serve to reduce transmittance.
Worst case scenario, you're out a few bucks, but still gain protection from any new scratches, and a fresh fingerprint coating to use.