Mean value of mixed exponential sums

For integers $q$, $m$, $n$, $k$ with $q,k\geq 1$, and Dirichlet character $\chi\bmod q$, we define a mixed exponential sum

$$C(m,n,k,\chi;q):= \mathop{{\sum}'}_{a=1}^q\chi(a)\hbox{e}\left(\frac{ma^k+na}{q}\right),$$

where $\hbox{e}(y)=\hbox{e}^{2\pi iy}$, and $\sum'_{a}$ denotes the summation over all $a$ with $(a,q)=1$. The main purpose of this paper is to study the mean value of

$$\sum_{\chi\bmod q}\mathop{{\sum}'}_{m=1}^q\left\vert C(m,n,k,\chi;q)\right\vert^4,$$

and to give a related identity on the mean value of the general Kloosterman sum

$$K(m,n,\chi;q):=\mathop{{\sum}'}_{a=1}^q\chi(a)\hbox{e}\left(\frac{ma +n\overline{a}}{q}\right),$$

where $a\overline{a} \equiv 1 \bmod q$.