  We give new decomposition theorems for classes of graphs that can be
transduced in first-order logic from classes of sparse graphs -- more
precisely, from classes of bounded expansion and from nowhere dense classes. In
both cases, the decomposition takes the form of a single colored rooted tree of
bounded depth where, in addition, there can be links between nodes that are not
related in the tree. The constraint is that the structure formed by the tree
and the links has to be sparse. Using the decomposition theorem for
transductions of nowhere dense classes, we show that they admit low-shrubdepth
covers of size $O(n^\varepsilon)$, where $n$ is the vertex count and
$\varepsilon>0$ is any fixed~real. This solves an open problem posed by
Gajarsk\'y et al. (ACM TOCL '20) and also by Bria\'nski et al. (SIDMA '21).
