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A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In the paper it is proved that a finite group $G$ is $p$-nilpotent provided $p$ is the smallest prime number dividing the order of $G$ and every minimal subgroup of $P\cap G^{\text{'}}$ is weakly-supplemented in $N_G\left(P\right),$ where $P$ is a Sylow $p$-subgroup of $G$. As applications, some interesting results with weakly-supplemented minimal subgroups of $P\cap G^{\text{'}}$ are obtained.