Let $G$ be a finite non-abelian $p$-group, where $p$ is an odd prime, $N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ is non-abelian. Does there exist an element $g\in G$ such that $\langle g\rangle$ is NOT normal in $G$ and $N\langle g\rangle$ is normal in $G$?(Note that $G$ and $\frac{G}{N}$ are non-Dedekind groups and contain non-normal cyclic subgroups. By a Dedekind group I mean a group all of whose subgroups are normal, which are abelian groups or direct product of a quaternion group, an elemntary abelian group and an abelian group with all elements of odd order (Hamiltonian group)).

Thank you very much!

Added later(According to professor Holt's answer): Let $G$ be a finite non-abelian $p$-group and $N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ be also non-abelian ($p$ is an odd prime). Also let for each $g\in G$ $$\langle g\rangle\lhd G\Leftrightarrow N\langle g\rangle\lhd G.$$ Is it possible to classify such $G$?