  We study the problem of finding pairwise vertex-disjoint copies of the
$\ell$-vertex cycle $C_\ell$ in the randomly perturbed graph model, which is
the union of a deterministic $n$-vertex graph $G$ and the binomial random graph
$G(n,p)$. For $\ell \ge 3$ we prove that asymptotically almost surely $G \cup
G(n,p)$ contains $\min \{\delta(G), \lfloor n/\ell \rfloor \}$ pairwise
vertex-disjoint cycles $C_\ell$, provided $p \ge C \log n/n$ for $C$
sufficiently large. Moreover, when $\delta(G) \ge\alpha n$ with $0<\alpha \le
1/\ell$ and $G$ is not `close' to the complete bipartite graph $K_{\alpha
n,(1-\alpha) n}$, then $p \ge C/n$ suffices to get the same conclusion. This
provides a stability version of our result. In particular, we conclude that $p
\ge C/n$ suffices when $\alpha>n/\ell$ for finding $\lfloor n/\ell \rfloor$
cycles $C_\ell$.
  Our results are asymptotically optimal. They can be seen as an interpolation
between the Johansson--Kahn--Vu Theorem for $C_\ell$-factors and the resolution
of the El-Zahar Conjecture for $C_\ell$-factors by Abbasi.
