Character sums over shifted smooth numbers

We give nontrivial bounds in various ranges for character sums of the form

$\sum_{n\in\mathcal S(x,y)}\chi(n +a), \qquad \gcd(a,p) = 1,$

where $\chi$ is a nontrivial multiplicative character modulo a prime $p$ and $\mathcal S(x,y)$ is the set of positive integers $n\le x$ that are divisible only by primes $q \le y$.