AbstractThis paper presents a measurement of the electroweak production of two jets in association with a $$Z\gamma $$ Z γ pair, with the Z boson decaying into two neutrinos. It also presents a search for invisible or partially invisible decays of a Higgs boson with a mass of 125 $$\text {GeV}$$ GeV produced through vector-boson fusion with a photon in the final state. These results use data from LHC proton–proton collisions at $$\sqrt{s}$$ s = 13 $$\text {TeV}$$ TeV collected with the ATLAS detector and corresponding to an integrated luminosity of 139 $$\hbox {fb}^{-1}$$ fb - 1 . The event signature, shared by all benchmark processes considered for the measurements and searches, is characterized by a significant amount of unbalanced transverse momentum and a photon in the final state, in addition to a pair of forward jets. Electroweak $$Z\gamma $$ Z γ production in association with two jets is observed in this final state with a significance of 5.2 (5.1 expected) standard deviations. The measured fiducial cross-section for this process is $$1.31\pm 0.29$$ 1.31 ± 0.29 fb. An observed (expected) upper limit of 0.37 ($$0.34^{+0.15}_{-0.10}$$ 0 . 34 - 0.10 + 0.15 ) at 95% confidence level is set on the branching ratio of a 125 $$\text {GeV}$$ GeV Higgs boson to invisible particles, assuming the Standard Model production cross-section. The signature is also interpreted in the context of decays of a Higgs boson into a photon and a dark photon. An observed (expected) 95% CL upper limit on the branching ratio for this decay is set at 0.018 ($$0.017^{+0.007}_{-0.005}$$ 0 . 017 - 0.005 + 0.007 ), assuming the Standard Model production cross-section for a 125 $$\text {GeV}$$ GeV Higgs boson.