I think that that player is too old to recognize all recently produced media, probably the burning speed is not the real issue. A firmware update for that player should solve, but I'm afraid that you'll not find any firmware update for it
Too bad, all recently produced media are manufactured to be burned at high speed, and if you burn these discs too slowly, you'll get a coaster.
The only reasonable solution is to get a more recent player. Another possibility is to get rid at all of optical supports, using a player able to read files from a flash memory like an USB pendrive or a secure digital card. Another solution could be connect a mp3 player to your player through a stereo connector for example, so you don't need at all any CD to be burned. If you are concerned about quality, you can avoid to compress your music to mp3 and save it as WAV files or FLAC (a lossless compression codec). Some players like the SanDisk Sansa Clip+ are able to read FLAC
Another (very difficult) solution is trying to find old media and burn these with an old burner, but there is a very low probability you can find old media and an old drive still working
Regarding media, I suggest to use only the very best, i.e. Verbatim and Taiyo Yuden: these are the only proven reliable media available in the market. All other brands have a hugely variable quality (from decent to ugly)