Hint: Let A be a decision procedure for HAM-CYCLE. Use A repeatedly on various modifications of the graph to determine your cycle.
Hint: Show HAM-PATH ≤P HAM-CYCLE. Note that your reduction needs an `if and only if' proof something like this:
G,u,v is in HAM_PATH if and only if G' (built out of G,u, and v) is in HAM_CYCLE. (where, of course, you have explained how G' is to be built)