This paper examines the evaluations of sixth power mean values of a generalized two-term exponential sums. In the case $p\equiv 3mod4$, we try to establish two precise formulas by applying the properties of character sums and the number of the solutions of relevant congruence equations modulo an odd prime $p$.

This paper mainly presents a new class of generalized Gauss sums. In order to find the potential relationship between these sums, we start from their definition, and study the fourth means of this new generalized Gauss sums by applying elementary analytic methods and some important qualities related to trigonometric sums. This result extends the classical Gauss sums and Kloosterman sums in analytic number theory, and makes a new contribution to the further development of this field. Finally, by calculating, an important identity is given. At the same time, we also put forward some problems that still exist and have not been solved in this field.