Singlino-dominated dark matter properties are investigated in the Z3 next-to-minimal supersymmetric Standard Model, producing superweak interactions with nucleons involved in dark matter direct-detection experiments. Approximate analytical formulas describing the dark matter abundance and cross section in the scattering with nucleons are used to illustrate a dependence on theoretical parameters in the neutralino and Higgs sectors. It is shown that the measured abundance requires a sizable singlet-doublet Higgs coupling parameter λ , while the experimental detection results prefer a small λ . The parameter space is then surveyed using a nest sampling technique guided by a likelihood function containing various observables in dark matter, Higgs, and B physics, such as the abundance and the scattering cross section. It is demonstrated that dark matter can achieve the correct abundance through χ∼10χ∼10→t t ¯ or coannihilation with Higgsinos. The former process provides significantly larger Bayesian evidence than the latter, but this will be examined by the near-future PandaX-4T experiment. If the experiment shows no signs of dark matter, it will become highly disfavored. Furthermore, four cases are summarized to suppress dark matter scattering with nucleons, namely, a small λ and three kinds of cancellation between different contributions.