Hmmm ... I don't know enough to be sure. It could be a number of things, I suppose.
On my wife's care, she was in the habit of twisting the power cord from her cellular phone. This caused the rear nut to back out and then fall off over time. PITA to take the dash out to get to it.
In my car, my radar detector "stopped working" on me some time back. Turned out that the pin in the center of the plug had fallen off (it turns to allow access to the fuse inside the plug). When I put the fuse and pin back in, it started working again!
My guess (and it is a guess) is that the fuse in your plug or in-line with the MP3 player is blown. Does the player work in another car? If not, then that is the culprit (fuse or otherwise). If it is working in another car, then the fuse to the socket in your car may be the problem. You would need to look into your manual to find the fusebox and check out the one for the cigarette lighter socket.
You may also have corrosion on the tip of the plug (or in the socket where it contacts the pin). This is typical when a spark occurs when plugging in and out. The metal gets a layer of burnt oxide that prevents current flow - try cleaning the pin and the socket.
Z