Let $\chi(n)$ be a quadratic character modulo a prime $p$. For a fixed integer $s\ne 0$, we estimate certain exponential sums with truncated $L$-functions \[L_{s,p}(n) = \sum_{j=1}^n \frac{\chi(\,j)}{j^s}\qquad (n =1, 2, \ldots)\]. Our estimate implies certain uniformly of distribution properties of reductions of $L_{s,p}(n)$ in the residue classes modulo $p$.