I find myself needing a good estiamate for $\sum_{p\le x,\, p\equiv a\mod q} 1/p$, perhaps something like $$ \sum_{p\le x,\, p\equiv a\mod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\exp(-c\sqrt{\log x})\big) $$ for $q$ up to a power of $\log x$, or something of that shape. Moreover, I'd like to be able to apply this uniformly in $q$, so the dependence of the error term on $q$ needs to be explicit (and the constant $b(q,a)$ needs to be explicit enough to manage as well).

Does anyone have a good reference to recommend for such sums?