We discuss a novel mechanism for generating masses for right-handed Majorana neutrinos, that goes beyond the conventional seesaw. The mechanism involves quantum fluctuations of a massless Kalb-Ramond (KR) pseudoscalar (axion-like) field, which exists in string-inspired extensions of the standard model. We assume a kinetic mixing of the KR field with ordinary (massive in general) axions, which also exist in string models, and which are assumed to couple to Majorana (right-handed) neutrinos via non-perturbatively-generated chirality changing Yukawa couplings, breaking the axion-shift symmetry. No vacuum expectation value is assumed for the axion fields. Majorana masses for the right-handed neutrinos are generated radiatively as a result of anomalous higher-loop axion-neutrino couplings. Implications for the Dark sector of the Universe are discussed. In particular, we explore the possibility of generating masses for the right-handed neutrinos of order of a few tens of keV. Such neutrinos could play an important role in the galactic structure, and more generally they could serve as a warm dark matter component in the Universe, providing a potential resolution to the so-called small-scale cosmology ``crisis'', that is, discrepancies between observations at galactic scales and numerical simulations based on the Lambda-CDM model. If such scenarios are realised in nature, they might imply that Dark Matter consists of more than one species (warm and cold), with distinct roles in the structure and evolution of the Universe: the cold one being still responsible for the large scale structure of the Universe, in accordance to the predictions of the Lambda-CDM model, which agree with a plethora of cosmological observations, but the warm component (keV sterile neutrino) playing a crucial role in the (observed) galactic structure.