Let h, b, c, and q ≥ 3 be integers, and let \(\overline{a }\) satisfy the equation \(a\overline{a }\equiv 1\left({\text{mod}} q\right).\) The general mth Cochrane sum is defined as \({C}_{m}\left(h,c,q\right)={\sum }_{a=1}^{{\prime}q}\left(\left(c\overline{a }/q\right)\right)\left(\left({ha}^{m}/q\right)\right).\) The purpose of this paper is to study the hybrid mean value involving hyper-Kloosterman sums and the mth Cochrane sum \({\sum }_{x=1}^{{\prime}q}I\left(m+1,h;q\right){C}_{m}\left(h,c,q\right)\) by applying the properties of Gauss sums, primitive characters, and a mean value theorem of Dirichlet L-functions. Similarly, we also have the hybrid mean value involving mth Cochrane sum and the general Kloosterman sum defined by Ye [Y.B. Ye, Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith., 86(3):255–267, 1998] as \({K}_{m}\left(b;c;q\right):={\sum }_{x=1}^{{\prime}q}e\left(\left({bx}^{m}+c\overline{x }\right)/q\right).\) For m = 1, we obtain a better asymptotic formula than that of Zhang in [W.P. Zhang, On a Cochrane sum and its hybrid mean value formula, J. Math. Anal. Appl., 267(1):89–96, 2002].
