I found in a few papers (eg. here: Burgess) a note which states
It is a well-known consequence of the theorem, due to Weil [...] then $\sum_{x=N+1}^{N+H} \chi(f(x)) \ll p^{1/2}\log{p}$
However, I couldn't find neither what are the restrictions on $N$ and $H$, nor what is the exact value of the best known constant for polynomials of degree $d$. Since $\sum_{x=0}^{p-1} \left( \frac{x^2 + 1}{p} \right) = -1$, there must me some restrictions on $N$ and $H$, because otherwise we could sum from $0$ to $Kp$ and get in result $-K$ for arbitrary huge $K$, which would contradict with the theorem. And unfortunately I don't know french (and to be honest my knowledge about algebra is too minuscule) to understand the Weil's original paper.
So my question is
What are the constraints on $N$ and $H$ and what is the best known constant for polynomials of degree $d$?