In a joint work with N. Wickramasekera (Cambridge) we develop a regularity and compactness theory for a class of codimension-1 integral n-varifolds with generalised mean curvature in L^{p}_{loc} for some p > n. Subject to suitable variational hypotheses on the regular part (namely stationarity and stability for the area functional with respect to variations that preserve the "enclosed volume") and two necessary structural assumptions, we show that the varifolds under consideration are "smooth" (and have constant mean curvature in the classical sense) away from a closed singular set of codimension 7. In the case that the mean curvature is non-zero, the smoothness is to be understood in a generalised sense, i.e. also allowing the tangential touching of two smooth CMC hypersurfaces (e.g. two spheres touching).