Old English has "scratch cover" that comes in light (yellow ish and medium (red ish) and dark (brown ish) . It will not "fix" the scratches but can make them less obvious by darkening the wood exposed by the scratch. The dark, applied in small amounts w/multiple applications, worked really well on my china cabinet but it is a natural wood finish rather than a painted surface.

Have not tried the car products that supposedly "erase" scratches but do know someone that used fine steel wool with circular motions to "erase" minor scratches on a white car. Not sure that would help with a wood surface or a dark color though.

Answering this to bump it for the pros to help you.

Paint is paint. Even if it’s a factory CV. Just wet sand it with some 2000 carbide. NBD.

I think sanding is probably your only hope