  Since both $W$-mass and muon $g-2$ can be affected by the mass splittings
among extra Higgs bosons $(H,~A,~H^\pm)$ in a 2HDM, we take a model with
$\mu$-$\tau$ LFV interactions to examine the two anomalies reported
respectively by CDF II and FNAL. We obtain the following observations: (i)
Combined with theoretical constraints, the CDF $W$-mass measurement disfavors
$H$ or $A$ to degenerate in mass with $H^\pm$, but allows $H$ and $A$ to
degenerate. The mass splitting between $H^\pm$ and $H/A$ is required to be
larger than 10 GeV. The $m_{H^\pm}$ and $m_{A}$ are favored to be smaller than
650 GeV for $m_H<120$ GeV, and allowed to have more large values with
increasing of $m_H$. (ii) After imposing other relevant experimental
constraints, there are parameter spaces that simultaneously satisfy (at
$2\sigma$ level) the CDF $W$-mass, the FNAL muon $g-2$ and the data of lepton
universality in $\tau$ decays, but the mass splittings among extra Higgs bosons
are strictly constrained.
