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Seems like Cloud Player, recently introduced to listen music from clouds, doesn’t work on Safari Browser in any of the iOS devices. Whatever be the issue if you want to listen to songs from Cloud Player on your iPad / iPhone / iPod, just download the file using the browser and you can still listen. This is not a solution but till Amazon gets some fix you have something in hand.
You can also select multiple files and then click on the menu right next to song to download. It will play the songs one after the other. Thanks to Gian Posatiere comment on TechCrunch Post.