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