Is #CYCLE #P-complete?
This is one of the problems (very briefly) discussed by Valiant, "The Complexity of Enumeration and Reliability Problems", SIAM J. Comput. 1979, doi:10.1137/0208032. See the mention of "elementary cycles" near the top of p. 417. Valiant gives a Turing reduction from the number of Hamiltonian cycles.