In this paper we have estimated one multi-variable character sum Sigma(p-2)(a=2) Sigma(p-1)(b=1) Sigma(p-2)(c=2) Sigma(p-1)(d=1) (a(2) - b(2)/p) (b(2) - 1/p) (c(2) - d(2)/p) (d(2) - 1/p) (a(2)c(2) - 1/p), for odd prime p. With the help of our estimate of the above character sum, we have studied the tenth power mean value of generalized quadratic Gauss sums using estimates for character sums and analytic methods.
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