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The semi-constrained NMSSM (scNMSSM), or NMSSM with non-universal Higgsmasses, can naturally predict a light dark matter under current constraintsincluding Higgs data, sparticle-mass bounds, dark matter searches, and muong-2, etc. In this work, we take this scenario of scNMSSM as an example to studythe funnel-annihilation mechanisms of light dark matter ($1\!\thicksim\!62$GeV) and the invisible Higgs decay. In this scenario we found that: (i) Therecan be four funnel-annihilation mechanisms for the LSP $\tilde{\chi}^0_1$,which are the $h_2$, $Z$, $h_1$ and $a_1$ funnel. (ii) For the $h_1$ and $a_1$funnel with right relic density, the $\tilde{\chi}^0_1$ mass is lighter than 12GeV, and the invisible Higgs decay can be $2\%$ at most. (iii) For the $h_2$and $Z$ funnel with right relic density, the invisible Higgs decay can be about$0.4\%$ and $1\%$ respectively at most. (iv) If the invisible Higgs decay wasdiscovered at the HL-LHC, the four funnel-annihilation mechanisms of light darkmatter may be all excluded with $\tilde{\chi}^0_1$ as the only dark mattersource. Four benchmark points, one for each mechanism, are proposed for futurechecking with updated experimental results.

Funnel annihilations of light dark matter and the invisible decay of the Higgs boson