It's possible that it's food debris, but it sounds more like tonsil stones, as Weffette suggested. Unlike food particles, these seem to be produced by the tonsils and can be very deeply imbedded. They have a very bad odor. Although they are made up of lots of bacteria, they aren't necessarily associated with a particular illness or infection. I really don't know if you can completely get rid of them without having your tonsils removed.
I'd guess that these white things are food particles that are getting stuck in the little holes in your tonsils. (I've had this same thing in the past). The reason they smell is because the food that is stuck is actually decaying.
Have you tried gargling with salt water for a few days? This should help shrink the tissue and push out the little particles, plus it would help with healing.
Those little white things are coming from your tonsils. Search the web for "tonsil debris" or "tonsil stones". I have seen info regarding these but can't remeber what they are called or what to do about them. Good luck!!