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The hMSSM is a special parameterization of the minimal supersymmetric extension of the Standard Model (MSSM) in which the mass of the lightest Higgs boson is automatically set to the LHC measured value, $M_h\!\!=\!\! 125$\,GeV, by adjusting the supersymmetric particle spectrum such that it provides the required amount of radiative corrections to the Higgs boson masses.\ The latter spectrum was in general assumed to be very heavy, as indicated by the present exclusion limits of the LHC, not to affect the phenomenology of the Higgs sector.\ In this work, we investigate the impact on the hMSSM by a light gaugino and higgsino sector, that is allowed by the present LHC data.\ In particular, we discuss the radiative corrections due to charginos and neutralinos to the Higgs boson masses and couplings and show that an hMSSM can still be realized in this context.\ We first describe how this scenario is implemented in the package SuSpect that generates the MSSM Higgs and supersymmetric spectra.\ We then analyze the possible impact of Higgs boson decays into these new states, as well as the reverse cascade channels with Higgs bosons in the final states, for the constraints on the MSSM Higgs sector at the LHC.\ We further explore the cosmological constraints on the hMSSM with a light gaugino--higgsino spectrum.\ We analyze the relic abundance of the lightest neutralino as a candidate of the dark matter in the Universe and the constraints on its mass and couplings by the present and future astroparticle physics experiments.