  In this study, we first define the local potential associated to a weakly
positive closed supercurrent in analogy to the one investigated by Ben Messaoud
and El Mir in the complex setting. Next, we study the definition and the
continuity of the $m$-superHessian operator for unbounded $m$-convex functions.
As an application, we generalize our previous work on Demailly-Lelong numbers
and several related results in the superformalism setting. Furthermore,
strongly inspired by the complex Hessian theory, we introduce the Cegrell-type
classes as well as a generalization of some $m$-potential results in the class
of $m$-convex functions.
