Let $G = \langle x,y,z \rangle$ be a $3$-generated group of order $p^6$ of exponent $p$ and class $2$. So $Z(P) = [P,P]=\Phi(P)$ is elementary abelian of order $p^3$, and so is $G/Z(P)$.

Let $N = \langle [x,y] \rangle$. So $N \lhd G$ with $|N|=p$. Then $Z(G/N) = Z(G)/N$, so for $g \in G$, we have $$\langle g \rangle \lhd G \Leftrightarrow g \in Z(G) \Leftrightarrow \langle g, N \rangle < Z(G/N) \Leftrightarrow \langle g, N \rangle \lhd G/N.$$

Let $G = \langle x,y,z \rangle$ be a $3$-generated group of order $p^6$ of exponent $p$ and class $2$. So $Z(P) = [P,P]=\Phi(P)$ is elementary abelian of order $p^3$, and so is $G/Z(P)$.

Let $N = \langle [x,y] \rangle$. So $N \lhd G$ with $|N|=p$. Then $Z(G/N) = Z(G)/N$, so for $g \in G$, we have $$\langle g \rangle \lhd G \Leftrightarrow g \in Z(G) \Leftrightarrow \langle g, N \rangle < Z(G/N) \Leftrightarrow \langle g, N \rangle \lhd G/N.$$

Added later: An easier type of example is $P = Q \times N$ where $Q$ is any nonabelian $p$-group and $N$ is any abelian $p$-group.

Let $G = \langle x,y,z \rangle$ be a $3$-generated group of order $p^6$ of exponent $p$ and class $2$. So $Z(P) = [P,P]=\Phi(P)$ is elementary abelian of order $p^3$, and so is $G/Z(P)$.

Let $N = \langle [x,y] \rangle$. So $N \lhd G$ with $|N|=3$. Then $Z(G/N) = Z(G)/N$, so for $g \in G$, we have $$\langle g \rangle \lhd G \Leftrightarrow g \in Z(G) \Leftrightarrow \langle g, N \rangle < Z(G/N) \Leftrightarrow \langle g, N \rangle \lhd G/N.$$

Let $G = \langle x,y,z \rangle$ be a $3$-generated group of order $p^6$ of exponent $p$ and class $2$. So $Z(P) = [P,P]=\Phi(P)$ is elementary abelian of order $p^3$, and so is $G/Z(P)$.

Let $N = \langle [x,y] \rangle$. So $N \lhd G$ with $|N|=p$. Then $Z(G/N) = Z(G)/N$, so for $g \in G$, we have $$\langle g \rangle \lhd G \Leftrightarrow g \in Z(G) \Leftrightarrow \langle g, N \rangle < Z(G/N) \Leftrightarrow \langle g, N \rangle \lhd G/N.$$