This paper is a follow up to an article by two of the authors dedicated to the study of Poincaré and logarithmic Sobolev inequalities for measures of the form $dμ= e^{-U} dν$ where $e^{-U}$ is seen as a perturbation of $dν$. Application to the same functional inequalities for convolution products are then discussed. In the present paper we investigate similar problems for weaker functional inequalities, namely weak Poincaré, weighted Poincaré, weak log-Sobolev and weighted log-Sobolev inequalities.