Finding a Hamiltonian cycle from perfect matching of a bipartite graph
If $G$ has a disjoint vertex cycle cover then I agree that $H$ must have a perfect matching, but I don't see the other direction (or I have misunderstood how you define $H$ exactly). I think the construction you were thinking of is the one described in this answer: https://cstheory.stackexchange.com/a/8570/38111.

As for your question (assuming that $H$ is now the bipartite graph constructed in the answer I linked), I think it depends on what you are asking exactly:


if you are asking "if there is a Hamiltonian cycle in $G$, then is there a perfect matching in $H$ that 'corresponds' to this cycle" then yes, since, as David Eppstein points out in his answer, the perfect matchings of $H$ correspond 1-for-1 with cycle covers of $G$, and a Hamiltonian cycle is in particular a vertex disjoint cycle cover.
if you are asking "can we use the same kind of trick to efficiently find such a Hamiltonian cycle", then probably no, since finding a Hamiltonian cycle is NP-hard.