  The objective of this paper is to obtain asymptotic results for shifted sums
of multiplicative functions of the form $g \ast 1$, where the function $g$
satisfies the Ramanujan conjecture and has conjectured upper bounds on square
moments of its L-function. We establish that for $H$ within the range
$X^{1-2\varepsilon} \leq H \leq X^{1-\varepsilon}$, there exist constants
$B_{f,h}$ such that $$ \sum_{X\leq n \leq 2X}
f(n)f(n+h)-B_{f,h}X=O_{f,\varepsilon}\big(X^{1-\varepsilon/2}\big)$$ for all
but $O_{f,\varepsilon}\big(HX^{-1/44+6\varepsilon+o(1)}\big)$ integers $h \in
[1,H].$ Additionally, we establish power-saving error terms for almost all
integers $h$ in $[1, H]$ when $X^{87/88+7\varepsilon} \leq H \leq
X^{1-\varepsilon}$ and apply them to deduce mean-value estimates of long
Dirichlet polynomials. Our method is based on the Hardy-Littlewood circle
method. In order to treat minor arcs, we use the convolution structure and a
cancellation of $g(n)$ that are additively twisted, applying some arguments
from a paper of Matom\"a{}ki, Radziwi\l{}\l{} and Tao.
