Let 𝑝 be a prime, 𝜒 denote the Dirichlet character modulo $p,f(x)=a_0+a_1 x+\cdots+a_kx^k$ is a 𝑘-degree polynomial with integral coefficients such that $(p, a_0,a_1,\ldots,a_k)=1$, for any integer 𝑚, we study the asymptotic property of

\begin{equation*}\sum\limits_{\chi\neq \chi_0}\left| \sum\limits^{p-1}_{a=1}\chi(a)e\left( \frac{f(a)}{p}\right)\right|^2 |L(1,\chi)|^{2m},\end{equation*}

where $e(y)=e^{2\pi iy}$. The main purpose is to use the analytic method to study the $2m$-th power mean of Dirichlet 𝐿-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.