For integer \(n\geqslant 1\) and real \(u\), let \(\Delta(n,u):=|\{d:d\mid n, e^u<d\leqslant e^{u+1}\}|\). The Erdős-Hooley Delta-function is then defined by \(\Delta(n):=\max_{u\in\mathbb{R}}\Delta(n,u)\). In the paper under review, the authors study the normal order of the \(\Delta\)-function, and approximate the weighted sum \(S(x;g):=\sum_{n\leqslant x}g(n) \Delta(n)\) for any arithmetic function \(g\) belonging to the class \(\mathcal{M}_A(y,c,\eta)\), where for \(A>0\), \(y\geqslant 1\), \(c>0\), \(\eta\in]0,1[\), consists of the arithmetic functions \(g\) that are multiplicative, non-negative, and satisfy the conditions (i) \(g(p^\nu)\leqslant A^\nu\) for \(\nu\geqslant 1\), (ii) \(g(n)\ll_\varepsilon n^\varepsilon\) for any \(\varepsilon>0\) and \(n\geqslant 1\), (iii) \(\sum_{p\leqslant x}{g(p)}=y\,\mathrm{li}(x)+O\big(x e^{-c(\log x)^\eta}\big)\) for any \(x\geqslant 2\). The authors prove that for any \(g\) in \(\mathcal{M}_A(y,c,\eta)\) with the above mentioned constants \(A, y, c, \eta\), and for any \(a>\sqrt{2}\log 2\) and \(x\geqslant 3\), \[ S(x;g)\ll x (\log x)^{2y-2}e^{a\sqrt{\log\log x}}. \] Also, regarding to the normal order of the \(\Delta\)-function, they prove that letting \(b>(\log 2)/(\log 2+1/\log 2-1)\), \[ \Delta(n)\leqslant (\log\log n)^{b}. \]