  In this paper, we develop a method of evaluating general exponential sums
with rational amplitude functions for multiple variables which complements
works by T. Cochrane and Z. Zheng on the single variable case. As an
application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic
formula for the (weighted) number of solutions of quadratic congruences of the
form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus
establishing an equidistribution result for these solutions.
