Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) \in F[x] \ : \ \deg(f(x)) \leq d \text{ and } f(0) = 0\}. $$ Note that $|\mathcal{P}_{\leq d}| = q^{d}$. For $f(x) \in F[x]$ let $$ S(f) = \sum_{y \in E} \psi(f(y)). $$
Do you know what are good bounds for the modulus of the average of the character sums $$ A_d = \dfrac{1}{q^{d}} \sum_{f(x) \in \mathcal{P}_{\leq d}} S(f)? $$ Of course the Weil bound may be applied here to each polynomial to get a bound. But I wonder if when we average over all such polynomials we can get better bounds than these?
Additionally, can we get better bounds if we restrict the monomials in each $f(x)$ above to induce permutations of $E$? Thanks.