I was wondering if the following could be established by the methods that go into e.g. Linnik:

$\textbf{Claim. } \text{Let $\chi$ be a nonprincipal quadratic character of conductor $q$, and (e.g.) $c := \frac{1}{100}$. Then $\sum_{p\leq q^c} (1+\chi(p))\geq q^\delta$ once $q\gg_\delta 1$.}$

Of course once $c\gg 1$ this follows from Linnik's theorem. I'm guessing the answer here is no, since the interval is so short, but I'm not sure since the required lower bound is fairly small. (Forgive me if I have overlooked a very easy obstruction!)

some(very small!) $C$. I wonder if just assuming the lower bound needed for Wolke is enough to get this quantitative form? ...Probably not, again. $\endgroup$