Kloosterman sums with multiplicative coefficients

We obtain several new bounds for sums of the form
$$ S_{q}(x;f)=\mathop{{\sum}'}_{n\le x}f(n)e_{q}(an^*+bn), $$
in which $q$ is a sufficiently large integer, $\sqrt{q}\,(\log{q})\ll x\le q$, $a$ and $b$ are integers with $(a,q)=1$, $e_{q}(v) = e^{2\pi iv/q}$, $f(n)$ is a multiplicative function satisfying certain conditions, $nn^*\equiv 1 \pmod{q}$, and the prime in the sum means that $(n,q)=1$. The results in this paper refine similar bounds obtained earlier by Gong and Jia.