  A graph $G$ is called universal for a family of graphs $\mathcal{F}$ if it
contains every element $F \in \mathcal{F}$ as a subgraph. Let
$\mathcal{F}(n,2)$ be the family of all graphs with maximum degree $2$. Ferber,
Kronenberg, and Luh [Optimal Threshold for a Random Graph to be 2-Universal, to
appear in Transactions of the American Mathematical Society] proved that there
exists a $C$ such that for $p \ge C (n^{-2/3} \log^{1/3} n )$ the random graph
$G(n,p)$ a.a.s is $\mathcal{F}(n,2)$-universal, which is asymptotically
optimal. For any $n$-vertex graph $G_\alpha$ with minimum degree
$\delta(G_\alpha) \ge \alpha n$ Aigner and Brandt [Embedding arbitrary graphs
of maximum degree two, Journal of the London Mathematical Society 48 (1993),
39-51] proved that $G_\alpha$ is $\mathcal{F}(n,2)$-universal for an optimal
$\alpha \ge 2/3$.
  In this note, we consider the model of randomly perturbed graphs, which is
the union $G_\alpha \cup G(n,p)$. We prove that $G_\alpha \cup G(n,p)$ is
a.a.s. $\mathcal{F}(n,2)$-universal provided that $\alpha>0$ and
$p=\omega(n^{-2/3})$. This is asymptotically optimal and improves on both
results from above in the respective parameter. Furthermore, this extends a
result of B\"ottcher, Montgomery, Parczyk, and Person [Embedding spanning
bounded degree subgraphs in randomly perturbed graphs, arXiv:1802.04603
(2018)], who embed a given $F \in \mathcal{F}(n,2)$ at these values. We also
prove variants with universality for the family $\mathcal{F}^\ell(n,2)$, all
graphs from $\mathcal{F}(n,2)$ with girth at least $\ell$. For example, there
exists an $\ell_0$ depending only on $\alpha$ such that for all $\ell \ge
\ell_0$ already $p=\omega(1/n)$ is sufficient for
$\mathcal{F}^\ell(n,2)$-universality.
