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... in particular you should look at Weil's Theorem (page 7) and Theorem $3^{\prime\prime}$ (page 16). In particular, the latter should give you a non-trivial estimate for your second sum as $\chi(F(x)) \overline{\chi(G(x))}$ can be rewritten as $\chi(F(x)/G(x))$ for all but a few values of $x$.
