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

Let $ q\ge3 $ 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\limits_{a = 1}^{q}e\left(\frac{ma^k+na^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.