Bombieri-Vinogradov theorem (taken from Wikipedia) states: Let $x$ and $Q$ be any two positive real numbers with $x^{1/2}\log^{-A}x\leq Q\leq x^{1/2}.$ Then $$\sum_{q\leq Q}\max_{y<x}\max_{1\le a\le q\atop (a,q)=1}\left|\psi(y;q,a)-{y\over\varphi(q)}\right|=O\left(x^{1/2}Q(\log x)^5\right).$$

Here $\varphi(q)$ is the Euler totient function, and $$\psi(x;q,a)=\sum_{n\le x\atop n\equiv a\bmod q}\Lambda(n),$$ where $\Lambda$ denotes the von Mangoldt function.

My question is: Is it possible to have the statement (slightly weaker statement ok too) with smaller $Q$ ? Would it be possible to have similar statements with $Q= X^{\delta}$ for some small $\delta > 0$ (possibly adding smooth weights if necessary)?