TLDR: I don't understand the bit in bold, i.e. where in the formula is the average q? Thanks.
The Bombieri-Vinogradov theorem states the following:
For any $A > 0$ there exists a $B = B(A)$ such that, for $Q= N^{\frac{1}{2}}(\log N)^{-B}$
$$\sum_{q \leq Q}\max_{\substack{a}\\{(a,q)=1}}|\psi(N;q,a) - \frac{N}{\varphi(q)}| \ll_{A} \frac{N}{(\log N)^{A}} $$
where $$\psi(N;q,a) = \sum_{\substack{n \leq N}\\ n \equiv a \mod q}\log n$$
if $n$ is a prime, and $0$ otherwise.
From my understanding, verbally this is addressing the error term of primes in arithmetic progression, for $\bf{q \leq Q}$ on average