  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.
