  We investigate an extension of the Minimal Supersymmetric Standard Model
(MSSM) containing a $SU(2)$ Higgs triplet of zero hypercharge and a gauge
singlet. We focus on a scenario of this model which allows a light pseudoscalar
and/or a scalar below $100$ GeV in the spectrum, consistent with the most
recent data from the LHC and the earlier data from the LEP experiments. The
triplet extension gives rise to two additional charged Higgs bosons which do
not couple to fermions but can decay into $Z W^\pm$. The presence of a very
light pseudoscalar opens $a_1 W^\pm$ decay mode for the light charged Higgs
boson just like $Z_3$ symmetric singlet-extension (NMSSM). A triplet type
charged Higgs can be produced via vector boson fusion at the tree-level which
is absent in 2HDM and MSSM and NMSSM. If such charged Higgs boson is pair
produced both $Z W^\pm$ and $a_1 W^\pm$ decay modes can be probed which can
prove the existence of triplet and singlet both.
