  For integers $k\ge 3$ and $1\le \ell\le k-1$, we prove that for any
$\alpha>0$, there exist $\epsilon>0$ and $C>0$ such that for sufficiently large
$n\in (k-\ell)\mathbb{N}$, the union of a $k$-uniform hypergraph with minimum
vertex degree $\alpha n^{k-1}$ and a binomial random $k$-uniform hypergraph
$\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-(k-\ell)-\epsilon}$ for $\ell\ge 2$ and
$p\ge C n^{-(k-1)}$ for $\ell=1$ on the same vertex set contains a Hamiltonian
$\ell$-cycle with high probability. Our result is best possible up to the
values of $\epsilon$ and $C$ and answers a question of Krivelevich, Kwan and
Sudakov.
