  For a monoid $M$ and a subsemigroup $S$ of the full transformation semigroup
$T_n$, the wreath product $M\wr S$ is defined to be the semidirect product
$M^n\rtimes S$, with the coordinatewise action of $S$ on $M^n$. The full wreath
product $M\wr T_n$ is isomorphic to the endomorphism monoid of the free $M$-act
on $n$ generators. Here, we are particularly interested in the case that
$S=Sing_n$ is the singular part of $T_n$, consisting of all non-invertible
transformations. Our main results are presentations for $M\wr Sing_n$ in terms
of certain natural generating sets, and we prove these via general results on
semidirect products and wreath products. We re-prove a classical result of
Bulman-Fleming that $M\wr Sing_n$ is idempotent generated if and only if the
set $M/L$ of $L$-classes of $M$ forms a chain under the usual ordering of
$L$-classes, and we give a presentation for $M\wr Sing_n$ in terms of
idempotent generators for such a monoid $M$. Among other results, we also give
estimates for the minimal size of a generating set for $M\wr Sing_n$, as well
as exact values in some cases (including the case that $M$ is finite and $M/L$
is a chain, in which case we also calculate the minimal size of an idempotent
generating set). As an application of our results, we obtain a presentation
(with idempotent generators) for the idempotent generated subsemigroup of the
endomorphism monoid of a uniform partition of a finite set.
