  In the NonHolomorphic Supersymmetric Standard Model (NHSSM), the Yukawa
couplings of the bottom quark ($y_b$) and the tau lepton ($y_{\tau}$) might
receive substantial supersymmetric (SUSY) radiative corrections which have
prominent dependencies on the NHSSM-specific trilinear soft parameters,
$A_b^\prime$ and $A_{\tau}^\prime$, respectively, in addition to their
well-known dependence on $\tan\beta$ as is already present in the Minimal SUSY
Standard Model (MSSM). We study to what extent these could affect the
production cross sections of the heavy Higgs bosons ($H$ and $A$) in
association with a pair of $b$-quarks and their decay branching ratios, in
particular, to a $\tau\bar{\tau}$ pair and compare them with those obtained in
the MSSM. Requiring compliance with the recently observed upper bounds on the
product of their total cross-section and the branching ratio to
$\tau\bar{\tau}$ at the 13 TeV run of the Large Hadron Collider (LHC), with
data worth 139 fb$^{-1}$, results in an altered exclusion region in the
customary $m_A-\tan\beta$ plane in the framework of the NHSSM when compared to
what is derived by the LHC experiments within an MSSM setup. Such alterations
are estimated to be pronounced only for large $\tan\beta$ ($\geq 40$) and for
large, negative $A_b^\prime$ when one finds a reinforced exclusion of the
$m_A-\tan\beta$ plane with an excluded $m_A$ value larger by $\approx 200$ GeV,
compared to the MSSM case, at $\tan\beta=60$. On the other hand, the maximum
relaxation in $m_A$, for a similarly large but positive $A_b^\prime$, barely
exceeds $\approx 100$ GeV as a result of complementary variations in production
cross-sections and decay branching fractions of the heavy, neutral Higgs
bosons.
