Counting cycles, and paths in graphs is a hard problem, see questions here, and related question for cycles for a given length $k$, here. The question is about the approximability of these graph problems, are they easily approximatable? Or do they have more fundamental implications for $NP$ complete problems, or #$P$ counting problems, hardness or otherwise. Any #$P$ hardness for approximately counting paths and cycles(from uniform distributions, or not) would imply, gaps between actually generating cycles paths(approximately) and enumerating the number of paths, and cycles. Also, what can be said about the $APX$ hardness implications under reasonable hardness reduction assumptions($L$-hardness, others)?
Approximately counting all paths (or cycles) in polynomial time implies NP=RP. There is a very simple reduction to amplify the weight of the longest path/cycle. See https://doi.org/10.1016/0304-3975(86)90174-X, section 5.