We investigate a concrete scenario of a light scalar with a mass around 1 MeV which can be connected to the origin of neutrino masses and simultaneously survive current bounds on relativistic degrees of freedom in the early universe. A feeble coupling to the Standard Model neutrinos can relax stringent bounds on the decays to photons inferred from the measured value of Neff. Interestingly, we find that such a scalar whose diphoton coupling is generated by a tree-level coupling to the muon of similar strength as that of the Standard Model Higgs boson can simultaneously explain the longstanding discrepancy in the measured value of the muon magnetic moment. We present a possible ultraviolet (UV) completion of this scenario providing a link between new physics in the early universe and the generation of neutrino masses.