Average first-passage times for character sums

Let $\epsilon >0$ and, for an odd prime $p$, set $$S_{\mathrm{\ell }}(p):=\sum _{n\le \mathrm{\ell }}\left(\frac{n}{p}\right)\mathrm{.}$$ Define the first-passage time $$f_{\epsilon }(p):=\min \{\mathrm{\ell }\ge 1:S_{\mathrm{\ell }}(p)<\epsilon \mathrm{\ell }\}\mathrm{.}$$ We prove that there exists a constant $c_{\epsilon }>0$ such that, as $x\to \mathrm{\infty }$, $$\sum _{p\le x}{f}_{\epsilon }(p)\sim c_{\epsilon }\frac{x}{\log x}\mathrm{.}$$