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 $s_0=\sigma_0+it_0$ with $\frac{1}{2}\leq\sigma_0<1$. 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 $L(1/2+it,f\otimes\chi)L(1/2+it,\chi)\neq0$ if the prime $q$ satisfies $q\gg (1+|t|)^{\frac{543}{25}+\varepsilon}$.