  The vacuum expectation value $v_s$ of a Higgs triplet field $\Delta$ carrying
two units of lepton number $L$ induces neutrino masses $\propto v_s$. The
neutral component of $\Delta$ gives rise to two Higgs particles, a pseudoscalar
$A$ and a scalar $S$. The most general renormalizable Higgs potential $V$ for
$\Delta $ and the Standard-Model Higgs doublet $\Phi$ 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 $Z\to A S$ 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 $\Delta$ dominantly
couples to the third-generation doublet $L_\tau=(\nu_\tau,\tau)$, the deficit
of $\nu_\tau$ 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.
