  Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in
$G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$
and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a
Sylow $p$-subgroup $P$ of $G$, we investigate the structure of $G$ under the
assumption that $N_G(P)$ is $p$-supersolvable or $p$-nilpotent and that certain
cyclic subgroups of $P$ with order $p$ or $4$ are $c$-embedded in $G$ with
respect to $P$. New characterizations of $p$-supersolvability and
$p$-nilpotence of finite groups will be obtained.
