  In this article, we extend our recent work on a Bombieri-Vinogradov-type
theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le
(\log x)^C$ to sparse sets of moduli $s\le x^{1/3-\varepsilon}$ with radical
rad$(s)\le x^{9/40}$. To derive our result, we combine our previous method with
a Bombieri-Vinogradov-type theorem for general moduli $s\le x^{9/40}$ obtained
by Roger Baker.
