  We introduce four supersymmetric (SUSY) axion models in which the strong CP
problem and the $\mu$ problem are solved with the help of the Peccei-Quinn
mechanism and the Kim-Nilles mechanism, respectively. The axion physics
enriches the SUSY model by introducing axion as a dark matter candidate and,
therefore, the lightest supersymmetric particle (LSP) could just be a part of
the total dark matter. For this reason, axion relieves the tensions between
SUSY models and numerous experimental measurements, such as the dark matter
direct detection experiments and the precise measurements of anomalous magnetic
moment of the muon $a_\mu$. In the present paper, we investigate the
constraints imposed by the latest $a_\mu$ measurements and LUX-ZEPLIN (LZ)
experiment on the relic density of the Higgsino-like LSP. Additionally, we
consider the constraints arising from the cosmology of saxions and axinos, and
their impacts on the parameter space of our models are carefully examined. For
the axion constituting the remaining portion of dark matter, we find that the
conventional misalignment mechanism can successfully account for the correct
dark matter relic density observed by the Planck satellite.
