LetXbe an (E,M)-category for sinks. A recently introduced notion of connectedness with respect to a closure operatorConXand to a class ofX-monomorphismsNis further analyzed. The notion ofN-connectedness hull of a class ofX-objects is introduced and a characterization of it is presented under the assumption ofNbeing closed under the formation of pullbacks. Moreover, a characterization of the related notion ofN-connectedness class is presented under the assumption thatXcontains a terminal object. Some examples are provided.