For $k\ge 2$ and $r\ge 1$ such that $k+r\ge 4$, we prove that, for any $α>0$, there exists $ε>0$ such that the union of an $n$-vertex $k$-graph with minimum codegree $\left(1-\binom{k+r-2}{k-1}^{-1}+α\right)n$ and a binomial random $k$-graph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-\binom{k+r-2}{k-1}^{-1}-ε}$ 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.