  For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs
$\mathcal{F}$, the relative Tur\'{a}n number $\mathrm{ex}(H,\mathcal{F})$ is
the maximum number of edges in an $\mathcal{F}$-free subgraph of $H$. In this
paper we give lower bounds on $\mathrm{ex}(H,\mathcal{F})$ for certain families
of hypergraph cycles $\mathcal{F}$ such as Berge cycles and loose cycles. In
particular, if $\mathcal{C}_\ell^3$ denotes the set of all $3$-uniform Berge
$\ell$-cycles and $H$ is a 3-uniform hypergraph with maximum degree $\Delta$,
we prove \[\mathrm{ex}(H,\mathcal{C}_4^{3})\ge \Delta^{-3/4-o(1)}e(H),\]
\[\mathrm{ex}(H,\mathcal{C}_5^{3})\ge \Delta^{-3/4-o(1)}e(H),\] and these
bounds are tight up to the $o(1)$ term.
