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, $q^{frac12+varepsilon}ll Nll q^A$. In this paper, we shall prove that $$ sum_{nleq N}f(n)chi(n+a)ll Nfrac{loglog q}{log q} $$ and that $$ sum_{nleq N}f(n)chi(n+a_1)cdotschi(n+a_t)ll Nfrac{loglog q}{log q}, $$ where $tgeq 2, a_1, ldots, a_t$ are distinct integers modulo $q$.
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