Fix minor scratches

[Beerpool]

I bought a XZ a month ago and since its so hard to find a good screen protector I've been taking it naked all around. Thing is... I've noticed some really minor screen scratches, unnoticeable by touch but noticable when using the phone under the sun. So Ive wanted to know if there is a WORKING method to fix it. It pisses me off that such a new and expensive phone is already scratched. Checked many sites but all of 'em say different things, while one recommends one method the other one tells never to do that so I wanted to ask on here. Also any good screen protectors while we're at it? Bought one 2 weeks ago on mediamarkt... But it didn't even stick well to the screen 

[Beerpool]

Example. Some wikihow tell to use toothpaste but other say not to use it since its abrasive and can damage gorilla glass. Found a method that involves light sanding and using a chemical compound but that sound either trolling or the worst idea ever for a first timer.