  To obtain a SM-like Higgs boson around 125 GeV in the Minimal Supersymmetric
Standard Model with minimal gauge mediation of supersymmetry breaking (GMSB), a
heavy stop at multi-TeV level is needed and incurs severe fine-tuning, which
can be ameliorated in the framework of the deformed GMSB with visible-hidden
direct Yukawa interactions (YGMSB). We examine some general features of the
YGMSB and focus on the scenario with Higgs-messenger couplings (H-YGMSB) which
can automatically maintain the minimal flavor violation (MFV). It turns out
that such a Yukawa mediation scenario can give a large -A_t and
-m_{stop_L,R}^2, leading to a maximal stop mixing, and thus can readily give a
light stop (stop_1) below the TeV scale. However, we find that in the minimal
H-YGMSB scenario, m_{H_u}^2 is too large and then the electroweak symmetry
breaking is inconsistent with the large stop mixing. To solve this problem, we
modify the hidden sectors in two ways, adding a new strong gauge dynamics or
introducing the (10,10_bar) messengers. For each case we present some numerical
study.
