There's this paint cleaner, Turtle wax scratch and swirl remover, see if it works for you.
Is that a plastic express mirror? If it is, you can try using a paint remover (thinner), apply a small amount to a cloth and rub it out gently. Do not apply too much pressure or you'll get color off the plastic. Then get another cloth to dry it.
But if the cover isn't plastic, DO NOT use thinner, it will ruin it.