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 pseduoscalar 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.