Unlike its minimal counterpart, the Next to Minimal supersymmetric Standard Model (NMSSM) allows the possibility that the lightest neutralino could have a mass as small as $sim 1 {rm GeV}$ while still providing a significant component of relic dark matter (DM). Such a neutralino can provide an invisible decay mode to the Higgs as well. Further, the observed SM-like Higgs boson ($H_{125}$) could also have an invisible branching fraction as high as $sim 19%$. Led by these facts, we first delineate the region of parameter space of the NMSSM with a light neutralino ($M_{{tilde{chi}}_{1}^{0}} < 62.5 {rm GeV}$) that yields a thermal neutralino relic density smaller than the measured relic density of cold dark matter, and is also compatible with constraints from collider searches, searches for dark matter, and from flavor physics. We then examine the prospects for probing the NMSSM with a light neutralino via direct DM detection searches, via invisible Higgs boson width experiments at future $e^+e^-$ colliders, via searches for a light singlet Higgs boson in $2b2mu$, $2b2tau$ and $2mu2tau$ channels and via pair production of winos or doublet higgsinos at the high luminosity LHC and its proposed energy upgrade. For this last-mentioned electroweakino search, we perform a detailed analysis to map out the projected reach in the $3l+{rm E{!!!/}_T}$ channel, assuming that chargino decays to $W {tilde{chi}}_{1}^{0}$ and the neutralino(s) decay to $Z$ or $H_{125}$ + ${tilde{chi}}_{1}^{0}$. We find that the HL-LHC can discover SUSY in just part of the parameter space in each of these channels, which together can probe almost the entire parameter space. The HE-LHC probes essentially the entire region with higgsinos (winos) lighter than 1 TeV (2 TeV) independently of how the neutralinos decay, and leads to significantly larger signal rates.
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