A search for final states characterised by at least one isolated high transverse-momentum photon, jets and large missing transverse momentum is presented. Such a final state might occur in gauge-mediated supersymmetric models where a pair of binos and higgsinos mix to form neutralinos, one of which decays into a photon plus a gravitino while the other decays into a Higgs boson, a Z boson or a photon, plus a gravitino. The search is performed using the full Run-2 data sample of 140 fb⁻¹ of s√=13 TeV proton-proton collisions collected by the ATLAS detector at the Large Hadron Collider. No significant excess of events is observed above the Standard Model prediction and model-dependent exclusion limits at the 95% confidence level are set. These limits are interpreted in terms of the masses of gauginos, which are excluded up to 1.2 TeV depending on their branching ratios, and on their branching ratios as a function of their mass.