Mixed incomplete character sums of rational functions with smooth moduli

Let $\chi ={\chi }_q$ be a primitive character mod $q$ and fix $\mathrm{\Delta }>0$. In 1989 Graham and Ringrose gave strong bounds on character sums ${\sum }_{M<n\le M+N}\chi (n)$ in intervals of length $N=q^{\mathrm{\Delta }}$ whenever $q$ is squarefree and is sufficiently smooth. Here we show that the smoothness parameter can be taken to be $N^{1-\epsilon }$. We also discuss various generalizations and applications, obtaining best possible results in several aspects.