  In this paper we have estimated one multi-variable character sum
\begin{align*}
\sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{c=2}^{p-2}\sum_{d=1}^{p-1}\left(\frac{a^2-b^2}{p}\right)\left(\frac{b^2-1}{p}\right)
  \left(\frac{c^2-d^2}{p}\right)\left(\frac{d^2-1}{p}\right)\left(\frac{a^2c^2-1}{p}\right),
\end{align*} 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.
