"Brasso" aka DuraGlit, Jewelers paste and T-Cut all work on essentially the same principle.
I restore antique watches and the lens is sometimes difficult / impossible to replace on pre 1950's watches (usually plastic of some sort on the really good ones) so as previous post, enjoy a mind numbing evening with a tin of duraglit !
Toothpaste and T-cut would work on a hard surface but the SPV has quite a soft plastic so I would (and have used myself) duraglit / jewelers paste (compund 3 or above only) will give a safer and clearer finish.
Dunno the wife swears blind this is the best smear free way to clean glass, but never tried it on a watch / mobile screen !
If it works it will probably take a long time.
Also as before, use a tiny amount, but no need to rub gently, as long as you do a circular motion and spread out beyond affected area.
As you are erasing part of the surface this will only work a number of times and depends how deep the scratchis. That said I have a 1960's Omega that I wear every day and have to duraglit the lens about once a month - been doing this for 5 + years and no obvious signs of problem !