Mixed incomplete character sums of rational functions with smooth moduli

Let $\chi=\chi_q$ be a primitive character mod $q$ and fix $\Delta>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} \chi(n)$ in intervals of length $N=q^\Delta$ 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.