A model-independent search for new physics leading to final states containing a Higgs boson, with a mass of 125.09 GeV, decaying to a pair of photons is performed with 139 fb^-1 of p root s = 13TeV pp collision data recorded by the ATLAS detector at the Large Hadron Collider at CERN. This search examines 22 final states categorized by the objects that are produced in association with the Higgs boson. These objects include isolated electrons or muons, hadronically decaying iota -leptons, additional photons, missing transverse momentum, and hadronic jets, as well as jets that are tagged as containing a b-hadron. No significant excesses above Standard Model expectations are observed and limits on the production cross section at 95% confidence level are set. Detector efficiencies are reported for all 22 signal regions, which can be used to convert detector-level cross-section limits reported in this paper to particle-level cross-section constraints.