  In this paper we study, in an open bounded set $\Omega\subset\mathbb R^N$
with Lipschitz boundary $\partial\Omega$, the Dirichlet problem for a nonlinear
singular elliptic equation involving the $1$--Laplacian and a total variation
term, that is, the inhomogeneous case of the equation appearing in the level
set formulation of the inverse mean curvature flow. Our aim is twofold. On the
one hand, we consider data belonging to the Marcinkiewicz space
$L^{N,\infty}(\Omega)$, which leads to unbounded solutions. So, we have to
begin introducing the suitable notion of unbounded solution to this problem.
Moreover, examples of explicit solutions are shown. On the other hand, this
equation allows us to deal with many related problems having a different
gradient term. It is known that the total variation term induces a regularizing
effect on existence, uniqueness and regularity. We focus on analyzing whether
those features remain true when general gradient terms are taken. Roughly
speaking, the bigger $g$, the better the properties of the solution.
