How to fix toilet jets clogged with chlorine tablet chunks
Bleach and chlorine can get nasty very fast if you try to put other chemicals in the tank to help the tablet dissolve.  I think the safest bet is to fill your tank with warm water and use that for several flushes.  Don't use water that is too hot because that could start to cause issues with the wax seal on the toilet if that gets too warm.  Just warm, touchable water should help dissolve it much faster than normal cold tank water.