The white chunks are simply food that is getting stuck in little holes on your tonsils. Food eventually decays and thats when you get the rotten taste in your mouth.
I know this because I used to have it for years (so did my mother and my sister).
The only solution I found was to "empty my tonsils" with a q-tip from time to time. After a while I became a pro at it.
I haven't had it for years though, and I didn't do anything besides gargling and good oral hygiene.