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 $\text {GeV}$ 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 $\text {TeV}$ collected with the ATLAS detector and corresponding to an integrated luminosity of 139 $\hbox {fb}^{-1}$. 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 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 ($0.34^{+0.15}_{-0.10}$) at 95% confidence level is set on the branching ratio of a 125 $\text {GeV}$ 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 ($0.017^{+0.007}_{-0.005}$), assuming the Standard Model production cross-section for a 125 $\text {GeV}$ Higgs boson.