Distribution of differences of characters evaluated at consecutive polynomial values

In this paper, we study the distribution of difference of multiplicative and additive characters modulo $p$ at consecutive polynomial values. More precisely, for an interval $I$ over finite field and $0<m<1$, we investigate the following sums \begin{align*} \sum_{n\in I}|\psi(F(n))-\psi(F(n+1))|^{2m} \quad \text{and} \quad \sum_{n\in I}|\chi(F(n))-\chi(F(n+1))|^{2m}, \end{align*} where $\psi$ is a non-trivial additive character and $\chi$ is a non-trivial multiplicative character modulo $p$, under suitable conditions on $\chi$ and $F$. As a consequence, we derive a formula for the first moment by specializing to $m=1/2$.