Shifted Character Sums with Multiplicative Coefficients, II

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{\frac12+\varepsilon}\ll N\ll q^A$. In this paper, we shall prove that $$\sum_{n\leq N}f(n)\chi(n+a)\ll N\frac{\log\log q}{\log q}$$ and that $$\sum_{n\leq N}f(n)\chi(n+a_1)\cdots\chi(n+a_t)\ll N\frac{\log\log q}{\log q},$$ where $t\geq 2, a_1, \ldots, a_t$ are distinct integers modulo $q$.