  The aim of this paper is to extend classic results of the theory of CMC
surfaces in the product spaces to the class of immersed surfaces in
$\mathbb{M}^2(\kappa)\times\mathbb{R}$ whose mean curvature is given as a $C^1$
function depending on their angle function. We cover topics such as the
existence of a priori curvature estimates for graphs, height estimates for
horizontal and vertical compact graphs and a structure-type result, which
classifies all the simply connected, properly embedded surfaces with finite
topology.
