Elliott has a result in this direction. Let $A>0$ and $a \geq 2$ a fixed integer. Furthermore, let $q \leq x^{1/3-\epsilon} $ be a large power of $a$. One then has that $$ \sum_{\substack{d \leq q^{-1}x^{1/2} \log^{-A-6}(x) \\ (d,q)=1 }} \max_{(r,qd)=1} \max_{y\leq x} \left|\pi(y,qd,r) - \frac{\text{Li}(y)}{\phi(qd)} \right| \ll_{A} \frac{x}{\phi(q) \log^{A}(x) } .$$

Elliott has a result in this direction. Let $A>0$ and $a \geq 2$ a fixed integer. Furthermore, let $q \leq x^{1/3-\epsilon} $ be a large power of $a$. One then has that $$ \sum_{\substack{d \leq q^{-1}x^{1/2} \log^{-A-6}(x) \\ (d,q)=1 }} \max_{(r,qd)=1} \max_{y\leq x} \left|\pi(y,qd,r) - \frac{\text{Li}(y)}{\phi(qd)} \right| \ll_{A} \frac{x}{\phi(q) \log^{A}(x) } .$$ Taking $d=1$ recovers the Bombieri-Vinogradov theorem. Note that while Bombieri-Vinogradov doesn't contain any information about specific arithmetic progressions with modulus larger than $\log^{A}(x)$, Elliott's inequality implies that $$\pi(x,q,r) \sim \frac{Li(x)}{\phi(q)}$$ for $q \leq x^{1/3-\epsilon}$ that is a large power of $a$.