On an error term for the first moment of twisted $L$-functions

Let $f$ be a Hecke-Maass cusp form for the full modular group and let $\chi$ be a primitive Dirichlet character modulo a prime $q$. Let $s_0={\sigma }_0+i{t}_0$ with $\frac{1}{2}\le {\sigma }_0<1$. We improve the error term for the first moment of $L(s_0,f\otimes \chi )\overline{L(s_0,\chi )}$ over the family of even primitive Dirichlet characters. As an application, we show that for any $t\in \mathbb{R}$, there exists a primitive Dirichlet character $\chi$ modulo $q$ for which $L(1\mathrm{/}2+it,f\otimes \chi )L(1\mathrm{/}2+it,\chi )\mathrm{\ne }0$ if the prime $q$ satisfies $q\gg (1+\mathrm{\mid }t\mathrm{\mid }{)}^{\frac{543}{25}+\epsilon }$.