Is Hamiltonian cycle fixed parameter tractable with parameter clique cover?
Yes, this problem is FPT. We showed this in the paper "Parameterized Edge Hamiltonicity" (Lampis, Makino, Mitsou, and Uno, DAM 2018).

In particular, in this paper we state the result for Hamiltonian Path, but things doesn't change much if you want to consider Hamiltonian Cycle. In the conference version (and the slightly outdated version on the arxiv) we state the result as parameterizing by the chromatic number of the complement of the input graph (which is equal to clique cover).