  We study Hamiltonicity and pancyclicity in the graph obtained as the union of
a deterministic $n$-vertex graph $H$ with $\delta(H)\geq\alpha n$ and a random
$d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular
graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $\alpha\in(0,1]$,
and also extend our result to a range of sublinear degrees. When $G$ is a
random $1$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all
$\alpha\in(\sqrt{2}-1,1]$, and this result is best possible. Furthermore, we
show that this bound on $\delta(H)$ is only needed when $H$ is `far' from
containing a perfect matching, as otherwise we can show results analogous to
those of random $2$-regular graphs. Our proofs provide polynomial-time
algorithms to find cycles of any length.
