Postnikov pieces and $B\mathbb{Z}/p$-homotopy theory

We present a constructive method to compute the cellularization with respect to $B^{m}\mathbb{Z}/p$ for any integer $m \geq 1$ of a large class of $H$-spaces, namely all those which have a finite number of non-trivial $B^{m}\mathbb{Z}/p$-homotopy groups (the pointed mapping space $\operatorname{map}_*(B^{m}\mathbb{Z}/p, X)$ is a Postnikov piece). We prove in particular that the $B^{m}\mathbb{Z}/p$-cellularization of an $H$-space having a finite number of $B^{m}\mathbb{Z}/p$-homotopy groups is a $p$-torsion Postnikov piece. Along the way, we characterize the $B\mathbb{Z}/p^r$-cellular classifying spaces of nilpotent groups.