  Very recently, a Fermilab report of muon $g-2$ showed a $4.2\sigma$
discrepancy between it and the standard model (SM) prediction. Motivated by
this inspiring result and the increasing tension in supersymmetric
interpretation of the anomalous magnetic moment, it is argued that in the
general next-to-minimal supersymmetric standard model (GNMSSM), a
singlino-dominated neutralino can act as a feasible dark matter (DM) candidate
in explaining the discrepancy naturally. In this case, the singlino-dominated
DM and singlet-dominated Higgs bosons can form a secluded DM sector with
$\tilde{\chi}_1^0\tilde{\chi}_1^0 \to h_s A_s$ responsible for the measured DM
relic abundance. when $m_{\tilde{\chi}_1^0} \gtrsim 150~{\rm GeV}$ and the
Yukawa coupling $\kappa$ is around $0.2$. This sector communicates with the SM
sector by weak singlet-doublet Higgs mixing, so the scatterings of the
singlino-dominated DM with nucleons are suppressed. Furthermore, due to the
singlet nature of the DM and the complex mass hierarchy, sparticle decay chains
in the GNMSSM are lengthened in comparison with the prediction of the minimal
supersymmetric standard model. These characteristics lead to sparticle
detection at the Large Hadron Collider (LHC) being rather tricky. This study
surveys a specific scenario of the GNMSSM, which extends the
$\mathbb{Z}_3$-NMSSM by adding an explicit $\mu$-term, to reveal the features.
It indicates that the theory can readily explain the discrepancy of the muon
anomalous magnetic moment without conflicting with the experimental results in
DM and Higgs physics, and the LHC searches for sparticles.
