I asked the Skillful Abbot about how he removed scratches on his Tamagotchis a while back. This was his response to me:
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Use fine, and ultrafine sanding sponges, wet.
Then use fine and ultra fine finishing polishes with a polishing cloth.
Then use automotive grade fine polishing creams.
Spend about an hour on each grade of sanding.
It works pretty well. But it isn't perfect. It doesn't take out really deep scratches, but smoothes over the finer scratches.
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I suspect that unless you already have some of the materials - sanding sponges, polishing creams, etc. to purchase them might be more than the cost of a new Tamagotchi. Also, if you don't have any experience using these, you may make the screen worse instead of better.