Character sums with exponential functions over smooth numbers

We give nontrivial bounds in various ranges for exponential sums of the form∑n∈S(x,y)exp⁡(2πiaϑn/m)and∑n∈St(x,y)exp⁡(2πiaϑn/m)where m ⩾ 2, φ is an element of order t in the multiplicative group Z*m, gcd(a,m) = 1, S(x,y) is the set of y-smooth integers n⩽x, and St(x,y) is the subset of S(x,y) consisting of integers that are coprime to t. We obtain sharper bounds in the special case that m = q is a prime number.