finite groups whose minimal subgroups are weakly $mathcal{h}^{ast}$-subgroups

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal{h}$-subgroup in‎ ‎$g$ if $n_g(h)cap h^{g}leq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ ‎$mathcal{h}^{ast}$-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal{h}$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assumption that every cyclic subgroup of $g$ of prime order ‎$p$ or of order $4$ (if $p=2$) is a weakly ‎$mathcal{h}^{ast}$-subgroup in $g$. our results improve‎ ‎and extend a series of recent results in the literature‎.