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 multiTeV 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 2 t L,R , leading to a maximal stop mixing, and thus can readily give a light stop (t 1 ) below the TeV scale. However, we find that in the minimal H−YGMSB scenario, m 2 Hu 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) messengers.For each case we present some numerical study.