If $N,H$ are powerfully embedded in $G$, I can prove that $NH$ is powerfully embedded in $G$. Therefore, $NH$ is powerful. Also, if $H$ is powerful and finitely generated, I can prove too that $NH$ is powerful. More generally,
if $H$ is powerful, can I prove that $NH$ is powerful?
I'm trying to show that $[NH,NH] \leq (NH)^{p}$.
My hypothesis are: $[H,H] \leq H^{p}$ and $[N,G] \leq N^{p}$. I work with the obvius approach: try to show $[NH,NH] \leq [H,H][N,G]$, but I don't think that works (at least, I cannot see how). So, I don't know which approach to follow.
I don't want a complete answer, just a hint.
Let $N \unlhd G$, and $H \leq G$, of the $p$-group $G$. The key is to observe that $\color{blue}{[NH,NH]=N'H'[N,H]}$. To prove this (sketch): observe that $[N,H] \unlhd \langle N,H \rangle = NH$. It follows that $[N,H] \unlhd [NH,NH]$. Working in the quotient mod $[N,H]$, where $\overline{N}$ and $\overline{H}$ commute one gets $\overline{[NH,NH]}=[\overline{NH},\overline{NH}]=[\overline{N},\overline{N}].[\overline{H},\overline{H}]=\overline{[N,N]}.\overline{[H,H]}$ and the formula in blue follows.
So, if $N$ is powerfully embedded and $H$ is powerful, then $N' \subseteq [N,G] \subseteq N^p$, and $H' \subseteq H^p$. Since $[N,H] \subseteq [N,G]$, the blue formula yields $[NH,NH] \subseteq N^pH^p \subseteq (NH)^p$, whence $NH$ is powerful.