Does anybody know the non-trivial bound for this sum?

$S(m,n,c,q)=\sum_{a,b\in \mathbb{Z}/cq\mathbb{Z}, \;ad\equiv 1\text{ mod }c} e^{2\pi i(am+nd)/qc},$

where $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$.

This seems to be not general bound but I want to know if there is conditions for $q$ for the Weil like bound to be valid, i.e., $|S(m,n,c,q)|\ll O(c^{1/2})$. Thanks.