The property ($gR$), introduced in [Aiena, P., Guillen, J. and Peña, P., {ı Property ($gR$) and perturbations}, to appear in Acta Sci. Math. (Szeged), 2012], is an extension to the context of B-Fredholm theory, of property ($R$), introduced in [Aiena, P., Guillen, J. and Peña, P., {ı Property ($R$) for bounded linear operators}, Mediterr. J. Math. {\bf 8} (4), 491-508, 2011]. In this paper we continue the study of property ($gR$) and we consider its preservation under perturbations by finite rank and nilpotent operators. We also prove that if $T$ is left polaroid (respṙight polaroid) and $N$ is a nilpotent operator which commutes with $T$ then $T+N$ is also left polaroid (respṙight polaroid).