  Let $k \geq 2$ be an integer. Kouider and Lonc proved that the vertex set of
every graph $G$ with $n \geq n_0(k)$ vertices and minimum degree at least $n/k$
can be covered by $k - 1$ cycles. Our main result states that for every $\alpha
> 0$ and $p = p(n) \in (0, 1]$, the same conclusion holds for graphs $G$ with
minimum degree $(1/k + \alpha)np$ that are sparse in the sense that \[
  e_G(X,Y) \leq p|X||Y| + o(np\sqrt{|X||Y|}/\log^3 n) \qquad \forall
X,Y\subseteq V(G). \] In particular, this allows us to determine the local
resilience of random and pseudorandom graphs with respect to having a vertex
cover by a fixed number of cycles. The proof uses a version of the absorbing
method in sparse expander graphs.
