It's almost impossible to find a CD player that won't play CD-Rs anymore. However, they are still around. Still, I'd have to say that the problem may not be your player, like everyone else said.
I also tried to burn a back up of the "cough cough" album. If you're referring to her first album, the problem is that it's an enhanced CD, and therefore, the songs are not recognized under the normal format. I've had this problem with all my enhanced CDs and I don't know what to do about it, so they just stay off the mix CDs.

Enhanced CD song files are seen as multimedia files instead.
The other things are that you may notice that some CDs aren't cut perfectly in between songs, sort of like (especially in rap CDs) the music is a little intermixed or there is talking that's starting to be blended it. Because there is no perfect blank gap of sound, they have to cut the song somewhere, and so although it may not sound like it listening to the whole CD through, if you have on song in a mix of them, you'll notice it's cut off. Or, there IS a gap, and then the funny noises are the beginning of the next song!
Just like Alec and AW said, the computer needs to devote ALL its power to burning. I usually turn off every application on my computer before running the burning program. The other thing would be that if you just got the burner, you may not have told the program to run it's diagnostics so that it can evaluate all of your CD drives to figure out the optimum reading characteristics, so that should help out too. The other thing is that it's difficult to do CD to CD copies. If the CD reader and burner have different reading and writing times, it forces the computer to coordinate both correctly. The best, albeit sort of slower way is to copy all the tracks to the harddrive and then burn the CD based on the files from the hard drive so it can take its sweet time.
That's my experience.
Anyway, if it is the Enhanced CD, I hope you've watched the video on it!
Jon