  We prove that the $k$-power of any planar graph $G$ is contained in
$H\boxtimes P\boxtimes K_{f(\Delta(G),k)}$ for some graph $H$ with treewidth
$15\,288\,899$, some path $P$, and some function $f$. This resolves an open
problem of Ossona de Mendez. In fact, we prove a more general result in terms
of shallow minors that implies similar results for many `beyond planar' graph
classes, without dependence on $\Delta(G)$. For example, we prove that every
$k$-planar graph is contained in $H\boxtimes P\boxtimes K_{f(k)}$ for some
graph $H$ with treewidth $15\,288\,899$ and some path $P$, and some function
$f$. This resolves an open problem of Dujmovi\'c, Morin and Wood. We generalise
all these results for graphs of bounded Euler genus, still with an absolute
bound on the treewidth.
  At the heart of our proof is the following new concept of independent
interest. An $\ell$-blocking partition of a graph $G$ is a partition of $V(G)$
into connected sets such that every path of length greater than $\ell$ in $G$
contains at least two vertices in one part. We prove that every graph of Euler
genus $g$ has a $894$-blocking partition with parts of size bounded by a
function of $\Delta(G)$ and $g$. Motivated by this result, we study blocking
partitions in their own right. We show that every graph has a $2$-blocking
partition with parts of size bounded by a function of $\Delta(G)$ and
$\textrm{tw}(G)$. On the other hand, we show that 4-regular graphs do not have
$\ell$-blocking partitions with bounded size parts.
