My guess would be that possibly your .m4a files date back to when iTunes was still using DRM? They no longer do that but any music bought when they did is convertable for an extra charge.
It won't play them if they are Apple Lossless. It should play lossy AACs, though. If not, you can get a music player that will in the Play store.
andLess is a simple free player that will play Apple Lossless m4as (as well as many other filetypes).
If you want to purchase a more rococco player, I would recommend PowerAmp.