The distribution of values of short hybrid exponential sums on curves over finite fields II

Let $p$ be a prime number, $C$ be any absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$ be rational functions. We continue the study of the distribution of the values of short hybrid exponential sums of the form $S_{H}(x;C) = \sum_{P\in C, x<x(P)\leq x+H}\chi(g(P))\psi(f(P))$ on $x\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions, the limiting distribution of the sum $S_{H}(x;C)$ is Gaussian for all curve $C$. This largely generalizes a previous result of the author and Zaharescu.