  The Higgs boson of 125 GeV requires large stop masses, leading to the large
$\mu$-parameter in most cases of gauge mediation. On the other hand, the
explanation for the muon $g-2$ anomaly needs small slepton and
neutralino/chargino masses. Such disparity in masses may be obtained from a
mass splitting of colored and non-colored messenger fields. However, even if
the required small slepton and neutralino/chargino masses are realized, all
parameter regions consistent with the muon g-2 are excluded by the recent
updated ATLAS result on the wino search in the case that the messenger fields
are in ${\bf 5}+\bar {\bf 5}$ representations of SU(5). It is also revealed
that the messenger fields in ${\bf 10} + \overline{\bf 10}$ or ${\bf 24}$
representation can not explain the muon g-2 anomaly. We show, giving a simple
example model, that the above confliction is solved if there is an additional
contribution to the Higgs soft mass which makes the $\mu$-parameter small. We
also show that the required Higgs B-term for the electroweak symmetry breaking
is consistently generated by radiative corrections from gaugino loops.
