We present an extension of a previously reported implementation of a Lanczos-driven coupled-cluster (CC) damped linear response approach to molecules in condensed phases, where the effects of a surrounding environment are incorporated by means of the polarizable embedding formalism. We are specifically motivated by a twofold aim: (i) computation of core excitations in realistic surroundings and (ii) examination of the effect of the differential response of the environment upon excitation solely related to the CC multipliers (herein denoted the J matrix) in computations of excitation energies and transition moments of polarizable-embedded molecules. Numerical calculations demonstrate that the differential polarization of the environment due to the first-order CC multipliers provides only minor contributions to the solvatochromic shift for all transitions considered. We thus complement previous works by confirming numerically the validity of the routinely invoked neglect of the J matrix contribution as well as motivating future use of the approximation that offers a reduction of the dimensionality of the eigenvalue problem. Preliminary applications to K-edge absorption of liquid water and aqueous acrolein are presented and highlight the importance of the environment that gives rise to transition-specific shifts. (c) 2014 AIP Publishing LLC.