  Searches for new particles beyond the Standard Model (SM) are an important
task for the Large Hadron Collider (LHC). In this paper, we investigate the
properties of the heavy non-SM Higgs bosons in the $\mu$-term extended
Next-to-Minimal Supersymmetric Standard Model ($\mu$NMSSM). We scan the
parameter space of the $\mu$NMSSM considering the basic constraints from Higgs
data, dark matter (DM) relic density, and LHC searches for sparticles. And we
also consider the constraints from the LZ2022 experiment and the muon anomaly
constraint at 2$\sigma$ level.
  We find that the LZ2022 experiment has a strict constraint on the parameter
space of the $\mu$NMSSM, and the limits from the DM-nucleon spin-independent
(SI) and spin-dependent (SD) cross-sections are complementary. Then we discuss
the exotic decay modes of heavy Higgs bosons decaying into SM-like Higgs boson.
We find that for doublet-dominated Higgs $h_3$ and $A_2$, the main exotic decay
channels are $h_3\rightarrow Z A_1$, $h_3\rightarrow h_1 h_2$, $A_2\rightarrow
A_1 h_1$ and $A_2\rightarrow Z h_2$, and the branching ratio can reach to about
23$\%$, 10$\%$, 35$\%$ and 10$\%$ respectively. At the 13 TeV LHC, the
production cross-section of $ggF\rightarrow h_3\rightarrow h_1 h_2$ and
$ggF\rightarrow A_2\rightarrow A_1 h_1$ can reach to about $10^{-11}$pb and
$10^{-10}$pb, respectively.
