Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k   be integers with q⩾3q⩾3 and k⩾1k⩾1, define the exponential sumS(m,n,k;q)=∑′a=1qe(ma+nakq), where e(x)=e2πixe(x)=e2πix, ∑′a∑′a denotes the summation over all a   such that (a,q)=1(a,q)=1. In this paper, we shall study the fourth power mean∑′n=1q|S(m,n,k;q)|4, and give some identities on the mean value of classical Kloosterman sums and hyper-Kloosterman sums.