  We study the decay of a standard model-like Higgs boson into a gravitino and
a neutralino, which subsequently decays promptly into another gravitino and a
photon. Such a decay can be important in scenarios where the supersymmetry
breaking scale is of the order of a few TeV, and in the region of low
transverse momenta of the photon, it may provide the dominant contribution to
the final state with a photon and two gravitinos. We estimate the relevant
standard model backgrounds and the prospects for discovering this Higgs decay
through a photon and missing transverse energy signal at the LHC in terms of a
simplified model. We also give an explicit model with manifest, but
spontaneously broken, supersymmetry in which the usual MSSM soft terms are
promoted to supersymmetric operators involving a dynamical goldstino
supermultiplet. This model can give rise to a SM-like CP-even neutral Higgs
particle with a mass of 125 GeV, without requiring substantial radiative
corrections, and with couplings sufficiently large for a signal discovery
through the above mentioned Higgs decay channel with the upcoming data from the
LHC.
