  Singlino-dominated dark matter properties are investigated in the $Z_3$
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 neutralino and Higgs sectors. It is shown that the measured
abundance requires a sizable singlet--doublet Higgs coupling parameter
$\lambda$, while the experimental detection results prefer a small $\lambda$.
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
$\tilde{\chi}_1^0 \tilde{\chi}_1^0 \to t \bar{t}$ or co-annihilation 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 $\lambda$ and three kinds of
cancellation between different contributions.
