  Type-X two Higgs doublet model is known to explain the muon $g-2$ anomaly
with a relatively light charged Higgs boson at large $\tan\beta$. The light
charged Higgs boson has been searched in the main $\tau \nu$ mode at the
colliders. Invoking a scenario of inverse seesaw as the origin of neutrino
masses and mixing, the charged Higgs boson can decay additionally to
right-handed neutrinos which leads to interesting phenomenology. Considering
generic lepton flavour violating signatures at the final states, a $5\sigma$
discovery can be achieved with the early data of LHC, at 14 TeV, for relatively
large inverse seesaw Yukawa coupling $Y_N$. The very light pseudoscalar and
charged Higgs boson mass reconstruction are performed using the new modes and
the results look promising. The inverse seesaw Yukawa coupling is shown to be
probed down to $Y_N \sim 0.2$ at HL LHC with 3000 fb$^{-1}$.
