finite groups whose minimal subgroups are weakly h*-subgroups

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ 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.