The distribution of powers of primes related to the Frobenius problem

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k\ge 1$, we prove that $$\mathrm{\#}\{p^k\le g_{c,d}:p\in \mathbb{P},p^k=cx+dy,x,y\in {\mathbb{Z}}_{⩾0}\}\sim \frac{k}{k+1}\frac{g^{1\mathrm{/}k}}{\log g}(\text{as }c\to \mathrm{\infty }),$$ which gives an extension of a recent result of Ding, Zhai and Zhao.