Do you have the latest version of iTunes? Also, do you have something else as the default player for M4A files? I've never heard of an M4A, but I was having trouble with MP4s.
M4A is just another name for AAC, which is what iTunes imports CDs in under as the default file type. I also have the latest version of iTunes and when I download a m4a file and double click on it, it automatically copies to iTunes and starts playing.