  We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $\mu>0$, there exist
$\alpha>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex
graph $G$ with $\delta(G)\geq \left(1-\frac{1}{r}+\mu\right)n$ and
$\alpha(G)\leq \alpha n$ contains an $r$-th power of a Hamilton cycle. We also
show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and
the $r=2$ case was recently conjectured by Staden and Treglown.
