Summary: Let \(m, n\) be integers such that \(1<m<n\). Let \(\mathcal{R}=M_n(\mathbb{D})\) be the ring of all \(n\times n\) matrices over a division ring \(\mathbb{D}\), \(\mathcal{M}\) an additive subgroup of \(\mathcal{R}\) and \(G:\mathcal{R}^m\rightarrow\mathcal{R}\) an \(m\)-additive map. In this paper, under a mild technical assumption, we prove that \(\delta_1(x)=G(x,\dots,x)\in\mathcal{M}\) for each rank-\(s\) matrix \(x\in\mathcal{R}\) implies \(\delta_1(x)\in\mathcal{M}\) for each \(x\in\mathcal{R}\), where \(s\) is a fixed integer such that \(m\leq s<n\), which has been considered for the case \(s=n\) in [Xu X, Zhu J., Central traces of multiadditive maps on invertible matrices, Linear Multilinear Algebra 2018; 66:1442-1448]. Also, an example is provided showing that the conclusion will not be true if \(s<m\). As applications, we also extend the conclusions by Liu, Franca et al., Lee et al. and Beidar et al., respectively, to the case of rank-\(s\) matrices for \(m\leq s<n\).