Powers of tight Hamilton cycles in randomly perturbed hypergraphs

For $k\ge 2$ and $r\ge 1$ such that $k+r\ge 4$, we prove that, for any $\alpha >0$, there exists $ϵ>0$ such that the union of an $n$-vertex $k$-graph with minimum codegree $(1-{\left(\genfrac{}{}{0px}{}{k+r-2}{k-1}\right)}^{-1}+\alpha )n$ and a binomial random $k$-graph ${\mathbb{G}}^{(k)}(n,p)$ with $p\ge n^{-\binom{k+r-2}{k-1}^{-1}-\epsilon}$onthesamevertexsetcontainsthe$r^{$\text{th}$}$ power of a tight Hamilton cycle with high probability. This result for $r=1$ was first proved by McDowell and Mycroft.