  We analyse the discovery prospects of the charged Higgs boson, $H^\pm$, via
its decay in the $W^\pm h$ channel in the minimal supersymmetric Standard Model
(MSSM) as well as several 2-Higgs Doublet Models (2HDMs). $h$, the lightest
scalar Higgs boson in these models, is identified with the recently discovered
$\sim 125$ GeV state, $H_\text{obs}$, at the Large Hadron Collider (LHC). We
find that, while it provides an important input in the kinematic selection of
signal events, the measured $H_\text{obs}$ mass renders this channel
inaccessible in the MSSM. In the 2HDMs though, through a dedicated
signal-to-background analysis for the $pp\to t(\bar{b})H^-\to \ell^\pm\nu_\ell
jj bb\bar{b}(\bar{b})$+h.c process, we establish that some parameter space
regions will be testable at the LHC with $\sqrt{s}=14$ TeV.
