This paper presents a measurement of the electroweak production of two jets in association with a $Z\gamma$ 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 GeV produced through vector-boson fusion with a photon in the final state. These results use data from LHC proton-proton collisions at $s$ = 13 TeV collected with the ATLAS detector and corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. 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 $Z\gamma$ 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 fb. An observed (expected) upper limit of 0.37 ($0.34-0.10+0.15$) at 95% confidence level is set on the branching ratio of a 125 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 ($0.017-0.005+0.007$), assuming the Standard Model production cross-section for a 125 GeV Higgs boson.