That doesn't make any sense to me. When you uploaded the MP3 file into your test area, I inserted it into the CuteEditor window instance there, saved it, checked "source," and it showed the full URL (see above) and would play. When I insert the same MP3 file into MY instance of CuteEditor, saved it, and checked the "Source," it shows an incorrect pathway and does not play.

If it's not possible to force full URL paths for media, and I did not enter a full URL when using either your test area or my own Cute Editor instance, what would explain the difference? There has to be a setting...