Let $$\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n)$$

where $\Lambda$ denotes the von Mangoldt function and $\phi$ to be Euler's totient function. Then the Siegel-Wafisz theorem states that given any real number $N$ there exists a positive constant $C_N$ depending only on $N$ such that

$$\psi(x;q,a)=\frac{x}{\varphi(q)}+O\left(x\exp\left(-C_N(\log x)^\frac{1}{2}\right)\right),$$

whenever $(a, q) = 1$ and $q\le(\log x)^N$.

Let $\omega$ be some smooth function, and consider $$ \sum_{n\leq x\atop n\equiv a\pmod q}\omega(n) \Lambda(n) $$ instead. Are there any results out there where we can obtain the result with a better range for $q$ by adding a smooth weigh to $\psi$? Thank you very much!