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Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. Subgroup $H$ is said to be weakly $ℳ$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G=HB$, and (2) if $H_1/H_G$ is a maximal subgroup of $H/H_G$, then $H_{1}B=B{H}_1<G$, where $H_G$ is the largest normal subgroup of $G$ contained in $H$. We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1<\left|D\right|<\left|P\right|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $\left|H\right|=\left|D\right|$ is weakly $ℳ$-supplemented in $G$. Some recent results are generalized.