Replace the spindle hub. All you need to do is dig up an old portable cd player(or make a trip to a thrift store),CAREFULLY pry the spindle hub off of the cd drive, and slide it onto the metal post on the psx cd drive after you remove the old spindle hub. I had that problem and now the cds dont skid.
I did it without the guide (I worked on my PSP Go before so I knew how to do it just based on my experience tearing it down and putting it back together) and now my PSP Go won't boot up.