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}$$forsomeshortinterval$$\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.