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 $\sqrt{s}$ = 13 TeV collected with the ATLAS detector and corresponding to an integrated luminosity of 139 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 $Z\gamma$ production in association with two jets is obseved 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) 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), assuming the Standard Model production cross-section for a 125 GeV Higgs boson.

The 95% CL upper limit on the Higgs boson production cross-section times branching ratio to $\gamma \gamma_\text{d}$ is shown for different VBF-produced scalar-mediator-mass hypotheses in the NWA. The theoretically predicted cross-section of a Higgs boson produced via VBF and with the $\mathcal{B}(H \to \gamma \gamma_\text{d}) =$ 5% is superimposed on the $\pm 1\sigma$ and $\pm 2\sigma$ NNLO QCD + NLO EW uncertainty band of the expected production cross-section limit.