Abstract Let $\lambda _{\phi }(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke–Maass cusp form on $\textrm{SL}_{2}(\mathbb Z)$ and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a nontrivial upper bound for the correlation $\sum _{n \leq X}f(n)\lambda _{\phi }(n+h)$ uniformly in $0<|h|\leq X$. As applications, we consider some special cases, including $f(n)=\lambda _{\pi }(n), \,\mu (n)\lambda _{\pi }(n)$ and any divisor-bounded multiplicative function. Here, $\lambda _{\pi }(n)$ denotes the $n$-th Dirichlet coefficient of a $\textrm{GL}_{m}$ automorphic $L$-function $L(s,\pi )$ for an irreducible cuspidal automorphic representation $\pi $, and $\mu (n)$ denotes the Möbius function. In particular, nontrivial savings are achieved for shifted convolution problems on $\textrm{GL}_{m}\times \textrm{GL}_{2}\, (m\geq 4)$ and Hypothesis C of Iwaniec–Luo–Sarnak for the first time.