  The updated measurements from the LHCb and SH0ES collaborations have
respectively strengthened the deviations of the ratio $R_{K}$ in rare
semi-leptonic $B$-meson decays and the present-day Hubble parameter $H_0$ in
the Universe, implying tantalizing hints of new physics beyond the Standard
Model. In this paper, we consider a simple flavor-specific two-Higgs-doublet
model, where the long-standing $R_{K^{(*)}}$ anomalies can be addressed by a
one-flavor right-handed neutrino. One of the intriguing predictions resulting
from the parameter space for the $R_{K^{(*)}}$ resolution under flavor- and
collider-physics constraints points toward a shift of the effective neutrino
number, $\Delta N_{\rm eff}=N_{\rm eff}-N_{\rm eff}^{\rm SM}$, as favored to
ease the $H_0$ tension. Depending on whether the neutrino is of Dirac or of
Majorana type, we show that the resulting shift is $\Delta N_{\rm eff}\simeq
1.0$ for the former and $\Delta N_{\rm eff}\simeq 0.5$ for the latter case,
respectively. While the Dirac case is disfavored by the CMB polarization
measurements, the Majorana solution is consistent with the recent studies via a
combined data set from various sources. Consequently, such a simple
flavor-specific two-Higgs-doublet model provides a link between the
$R_{K^{(*)}}$ anomalies and the $H_0$ tension, which in turn can be readily
verified or falsified by the upcoming measurements.
