A Bombieri-Vinogradov theorem for exponential sums over products of k primes

We prove a Bombieri-Vinogradov type theorem for exponential sums over products of $k$ primes. As an application, we show the lower bound $$\underset{n\le x}{\sup }\mid \sum _{m\le n}{1}_{\mathrm{\Omega }(m)=k}e(\alpha m)\mid \gg x^{1\mathrm{/}6-\epsilon }$$ for $2\le k\le (2-\epsilon )\log \log x$ and $\alpha \in \mathbb{R},$ where we noted $e(\beta ):=e^{2i\pi \beta }\mathrm{.}$