Hybrid mean value involving some two-term exponential sums and fourth Gauss sums

Let $q\ge 3$ be a positive integer. For any integers $m$, $n$, $k$, $h$, the two-term exponential sums $C(m,n,k,h;q)$ is defined as $C(m,n,k,h;q)=\sum _{a=1}^{q}e\left(\frac{m{a}^k+n{a}^h}{q}\right)$, where $k>h\ge 2$. The main purpose of this paper is to use analytic methods and the properties of classical Gauss sums to study the mean value involving two-term exponential sums and fourth Gauss sums, and to provide some asymptotic formulas and identities. Previously, only the case of $h=1$ had been studied.