Hamiltonicity in randomly perturbed hypergraphs

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