  In a recent work, Allen, B\"{o}ttcher, H\`{a}n, Kohayakawa, and Person
provided a first general analogue of the blow-up lemma applicable to sparse
(pseudo)random graphs thus generalising the classic tool of Koml\'{o}s,
S\'{a}rk\"{o}zy, and Szemer\'{e}di. Roughly speaking, they showed that with
high probability in the random graph $G_{n,p}$ for $p \geq C(\log
n/n)^{1/\Delta}$, sparse regular pairs behave similarly as complete bipartite
graphs with respect to embedding a spanning graph $H$ with $\Delta(H) \leq
\Delta$. However, this is typically only optimal when $\Delta \in \{2,3\}$ and
$H$ either contains a triangle ($\Delta = 2$) or many copies of $K_4$ ($\Delta
= 3$). We go beyond this barrier for the first time and present a sparse
blow-up lemma for cycles $C_{2k-1}, C_{2k}$, for all $k \geq 2$, and densities
$p \geq Cn^{-(k-1)/k}$, which is in a way best possible. As an application of
our blow-up lemma we fully resolve a question of Nenadov and \v{S}kori\'{c}
regarding resilience of cycle factors in sparse random graphs.
