I'm not sure what your fallen part is. But, for the smoking bit... You'd better sit down... ready?
The 77-78 model 750F models had a valve train like no other 750. The design was changed for more power, but consequently the valve guides would wear rapidly. No valve guide seal can last long with a wobbly valve stem going on. And the adjusters wear and damage both themselves and the valve stem end.
The "fix" by Honda was hardened valve guides (often done under warranty). But, it never did match the longevity of the earlier 750s and the K models of the same era.
The repair you are likely looking at is new valve guides, a valve job (seats at the very least). If the valve faces are still ok, then the stems have to be milled and lash caps installed to restore valve height, along with new tappet adjusters.
If you are strapped for money, you are probably crying by now. And, it is often cheaper to find a 77-78 K motor that will drop into your frame. This would also cure your clutch issues, what ever they are. You can paint the K engine black and ignore the F models higher redline mark on the tach. No one else needs to know.