  A search is presented for the pair production of higgsinos $\tilde{\chi}$ in
gauge-mediated supersymmetry models, where the lightest neutralinos
$\tilde{\chi}_1^0$ decay into a light gravitino $\tilde{G}$ either via a Higgs
$h$ or $Z$ boson. The search is performed with the ATLAS detector at the Large
Hadron Collider using 139 fb$^{-1}$ of proton-proton collisions at a
centre-of-mass energy of $\sqrt{s}$ = 13 TeV. It targets final states in which
a Higgs boson decays into a photon pair, while the other Higgs or $Z$ boson
decays into a $b\bar{b}$ pair, with missing transverse momentum associated with
the two gravitinos. Search regions dependent on the amount of missing
transverse momentum are defined by the requirements that the diphoton mass
should be consistent with the mass of the Higgs boson, and the $b\bar{b}$ mass
with the mass of the Higgs or $Z$ boson. The main backgrounds are estimated
with data-driven methods using the sidebands of the diphoton mass distribution.
No excesses beyond Standard Model expectations are observed and higgsinos with
masses up to 320 GeV are excluded, assuming a branching fraction of 100% for
$\tilde{\chi}_1^0\rightarrow h\tilde{G}$. This analysis excludes higgsinos with
masses of 130 GeV for branching fractions to $h\tilde{G}$ as low as 36%, thus
providing complementarity to previous ATLAS searches in final states with
multiple leptons or multiple $b$-jets, targeting different decays of the
electroweak bosons.
