  It is known that in the absence of a gauge singlet field, a specific class of
supersymmetry (SUSY) breaking non-holomorphic (NH) terms can be soft breaking
in nature so that they may be considered along with the Minimal Supersymmetric
Standard Model (MSSM) and beyond. There have been studies related to these
terms in minimal supergravity based models. Consideration of an F-type SUSY
breaking scenario in the hidden sector with two chiral superfields however
showed Planck scale suppression of such terms. In an unbiased point of view for
the sources of SUSY breaking, the NH terms in a phenomenological MSSM (pMSSM)
type of analysis showed a possibility of a large SUSY contribution to muon
$g-2$, a reasonable amount of corrections to the Higgs boson mass and a drastic
reduction of the electroweak fine-tuning for a higgsino dominated
$\widetilde{\chi}^0_1$ in some regions of parameter space. We first investigate
here the effects of the NH terms in a low scale SUSY breaking scenario. In our
analysis with minimal gauge mediated supersymmetry breaking (mGMSB) we probe
how far the results can be compared with the previous pMSSM plus NH terms based
study. We particularly analyze the Higgs, stop and the electroweakino sectors
focusing on a higgsino dominated $\widetilde{\chi}^0_1$ and
$\widetilde{\chi}^{\pm}_1$, a feature typically different from what appears in
mGMSB. The effect of a limited degree of RG evolutions and vanishing of the
trilinear coupling terms at the messenger scale can be overcome by choosing a
non-minimal GMSB scenario, such as one with a matter-messenger interaction.
