Step 1: Reduce to L-functions. By character orthogonality: $\psi(x; q, a) = \frac{1}{\phi(q)} \sum_{\chi \bmod q} \overline{\chi(a)} \psi(x, \chi)$ where $\psi(x, \chi) = \sum_{n \leq x} \chi(n)\Lambda(n)$.

Step 2: Explicit formula for each $\chi$. For each character $\chi$ mod $q$: $\psi(x, \chi) = \delta_{\chi = \chi_0} x - \sum_\rho \frac{x^\rho}{\rho} + O(\log^2(qx))$ where the sum is over nontrivial zeros of $L(s, \chi)$.

Step 3: Zero-free region. By the classical zero-free region: $L(s, \chi) \neq 0$ for $\sigma > 1 - c/\log(q(|t|+2))$, with at most one exceptional real zero $\beta_1$ (a Siegel zero) of a real character $\chi_1$.

Step 4: Bounding the zero sum. For non-exceptional zeros, $|x^\rho| = x^\beta \leq x^{1-c/\log(qT)}$ for $|\gamma| \leq T$. Summing over zeros (using the zero-counting formula $N(T, \chi) \ll T\log(qT)$) and choosing $T = \exp(\sqrt{\log x})$: the contribution is $O(x\exp(-c'\sqrt{\log x}))$ for $q \leq (\log x)^A$.

Step 5: Handling the Siegel zero. Siegel's theorem: for any $\varepsilon > 0$, $\beta_1 < 1 - c(\varepsilon)q^{-\varepsilon}$. For $q \leq (\log x)^A$ and $\varepsilon$ small enough: $x^{\beta_1} \leq x \cdot x^{-c(\varepsilon)q^{-\varepsilon}} \leq x \exp(-c(\varepsilon)(\log x)^{1-A\varepsilon})$. Choosing $\varepsilon = 1/(2A)$ makes this $\leq x\exp(-c''\sqrt{\log x})$.

The constant $c(\varepsilon)$ (hence $C(A)$) is ineffective because Siegel's lower bound for $L(1, \chi_1)$ is proved by contradiction: either there are few real zeros (and the bound is trivial) or a zero exists but forces $L(1, \chi_2) > 0$ for all other real characters via the Goldfeld-Gross-Zagier approach. $\square$