  We prove that for integers $2 \leq \ell < k$ and a small constant $c$, if a
$k$-uniform hypergraph with linear minimum codegree is randomly `perturbed' by
changing non-edges to edges independently at random with probability $p \geq
O(n^{-(k-\ell)-c})$, then with high probability the resulting $k$-uniform
hypergraph contains a Hamilton $\ell$-cycle. This complements a recent
analogous result for Hamilton $1$-cycles due to Krivelevich, Kwan and Sudakov,
and a comparable theorem in the graph case due to Bohman, Frieze and Martin.
