Recommend an MP3 CD player
Are you sure it's the CD player and not a problem with a batch of CD-Rs that you're using?
At any rate, I'm not sure what a recommendation on a portable MP3 CD player would look like, other than "don't buy the cheapest thing on the shelf". I have a 4yo RCA player that I paid about $25 for, right after MP3 CD players started to fall below $30, and it's always worked just fine. The reduced motor speed for MP3 playback means it will run MP3 discs for hours on end from a pair of AA batteries. (It also happened to have an auxiliary power jack that was compatible with the accessories from my then-dying Sony CarDiscman, which was a plus.)