The vacuum expectation value ${v}_{s}$ of a Higgs triplet field $\mathrm{\ensuremath{\Delta}}$ carrying two units of lepton number $L$ induces neutrino masses $\ensuremath{\propto}{v}_{s}$. The neutral component of $\mathrm{\ensuremath{\Delta}}$ gives rise to two Higgs particles, a pseudoscalar $A$ and a scalar $S$. The most general renormalizable Higgs potential $V$ for $\mathrm{\ensuremath{\Delta}}$ and the Standard-Model Higgs doublet $\mathrm{\ensuremath{\Phi}}$ does not permit the possibility that the mass of either $A$ or $S$ is small, of order ${v}_{s}$, while the other mass is heavy enough to forbid the decay $Z\ensuremath{\rightarrow}AS$ to comply with LEP 1 data. We present a model with additional dimension-6 terms in $V$, in which this feature is absent and either $A$ or $S$ can be chosen light. Subsequently we propose the model as a remedy to cosmological anomalies, namely the tension between observed and predicted tensor-to-scalar mode ratios in the cosmic microwave background and the different values of the Hubble constant measured at different cosmological scales. Furthermore, if $\mathrm{\ensuremath{\Delta}}$ dominantly couples to the third-generation doublet ${L}_{\ensuremath{\tau}}=({\ensuremath{\nu}}_{\ensuremath{\tau}},\ensuremath{\tau})$, the deficit of ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ events at IceCube can be explained. The singly and doubly charged triplet Higgs bosons are lighter than 280 GeV and 400 GeV, respectively, and could be found at the LHC.