Let $\mathbb F$ be a finite field of characteristic $p$, and size $q=p^n$. Let $\chi$ be a multiplicative character of order $m$. Then, it is well-known that the Weil sum is bounded as follows. $$W_{\mathbb F,d}(a) = \left| \sum_{x\in \mathbb F} \chi (x^d-ax) \right| \leq (d-1)\sqrt{q}$$
I'm wondering that whether there is such generic (or for specific $d$) nontrivial bound for $d > \sqrt{q}$. Or is this bound so tight that there is no such a nontrivial bound?
It is true that $(d−1)\sqrt q$ is an uninteresting bound when $d>\sqrt q$. This is because the sum has $q$ terms, each with absolute value one. So when $d−1\ge\sqrt{q}$, the trivial bound from the triangle inequality becomes better.
There are examples when the bound is tight. For example, if $n$ is even, and $\chi$ is a character that is trivial on the subfield $\Bbb{E}$ of size $\sqrt q$, then $\chi(x^{\sqrt q}+x)=1$ for all $x$ because $x^{\sqrt q}+x$ belongs to $\Bbb{E}$ (as the relative trace of $x$).
I'm sure there are also examples of high degree binomials such that a non-trivial bound can be found (so a specific $q,d$ and $a$).