Why is part of this Derailleur loose? (Sun Tour Cyclone II)
It is hard to give you the exact reasons why this is happening, but your derailleur body lost a good mechanic connection with the cage. Maybe the screw that holds them has just became loose, or maybe it has gone bad, or maybe your derailleur is just old. Hard to tell from the pictures.
I don't want to be pessimistic, but repair will be hard, as manuals and parts are hard to find, and I suspect even if you successfully do it there will still be too much slack for this part to perform adequately.
It sure doesn't hurt to try. The only link I can point you to is for an overhaul of a Shimano derailleur, here. These components aren't too different and this should give you a help.
In the meanwhile, I would start putting some money on the side for a new one, the good news are that if you can't fix yours you can get something really cheap. If you ride a lot, it is better to invest in something a little bit better.
Good luck!