Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$. By the Lang-Weil bounds $ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ depending on $n$, $d$ and $N$.

Are there any bounds on $C$? Can you improve the estimate in special situations? I am interested in the case of del Pezzo surfaces where I want to use to test for local solubility.