Character sums over smooth numbers

Let $\mathrm{\Psi }(x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We show that $$\frac{1}{\phi (q)}\sum _{\chi q}\mid \sum _{\begin{array}{c}n\le x\\ P(n)\le y\end{array}}\chi (n)\mid =o\left(\sqrt{\mathrm{\Psi }(x,y)}\right),$$ whenever $(\log x{)}^6\le y\le x^{\frac{1}{32\log \log x}}$ and $q\ge x^{1+\epsilon }$ for some small quantifiable $\epsilon >0$. The saving is substantial when $\epsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums.