Oh got it, you're using a flash based player, mobile devices do not have flash. The built-in logic for iOS is to display a play image that when clicked, opens the media in the media player on the device. If it is opening a new window to the media file, then my guess is that you have a JavaScript error in your page.
I am able to replicate the problem with Firefox, which indicates there is a JavaScript issue.
You may have better luck switching the audio player. Go to PowerPress menu on the left in your WordPress site, select "Audio Player", then click the link at the top to select a different player. Pick the top player listed, it is HtML5 based, which is the latest and greatest player.