  Recent discovery of the SM-like Higgs boson with $m_h\simeq 125$ GeV
motivates an extension of the minimal supersymmetric standard model (MSSM),
which involves a singlet Higgs superfield with a sizable Yukawa coupling to the
doublet Higgs superfields. We examine such singlet-extended SUSY models with a
Peccei-Quinn (PQ) symmetry that originates from an anomalous $U(1)_A$ gauge
symmetry. We focus on the specific scheme that the PQ symmetry is spontaneously
broken at an intermediate scale $v_{\rm PQ}\sim \sqrt{m_{\rm SUSY}M_{\rm Pl}}$
by an interplay between Planck scale suppressed operators and tachyonic soft
scalar mass $m_{\rm SUSY}\sim \sqrt{D_A}$ induced dominantly by the $U(1)_A$
$D$-term, $D_A$. This scheme also results in spontaneous SUSY breaking in the
PQ sector, generating the gaugino masses $M_{1/2}\sim \sqrt{D_A}$ when it is
transmitted to the MSSM sector by the conventional gauge mediation mechanism.
As a result, the MSSM soft parameters in this scheme are induced mostly by the
$U(1)_A$ $D$-term and the gauge mediated SUSY breaking from the PQ sector, so
that the sparticle masses can be near the present experimental bounds without
causing the SUSY flavor problem. The scheme is severely constrained by the
condition that a phenomenologically viable form of the low energy operators of
the singlet and doublet Higgs superfields is generated by the PQ breaking
sector in a way similar to the Kim-Nilles solution of the $\mu$ problem, and
the resulting Higgs mass parameters allow the electroweak symmetry breaking
with small $\tan\beta$. We find two minimal models with two singlet Higgs
superfields, satisfying this condition with a relatively simple form of the PQ
breaking sector, and briefly discuss some phenomenological aspects of the
model.
