Let $q \in \mathbb{N}$ and $\chi$ a Dirichlet character mod $q$. Let $F(x)$ be a polynomial with integer coefficients. I was wondering if a bound for the following sum was available or not: $$ \sum_{0 \leq x < q} \chi(F(x)). $$

I would appreciate any comments and references. Thank you very much!

PS In fact I am also interested in if a more general sum $$ \sum_{0 \leq x < q} \chi(F(x)) \overline{\chi(G(x) )} $$ can be bounded or not. Here $G$ is another polynomial with integer coefficients.

PPS For the first sum I am particularly interested in the case when $F(x)$ is not identically a constant modulo $q$.