Abstract This paper presents a measurement of the electroweak production of two jets in association with a $$Z\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> 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 $$\text {GeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>GeV</mml:mtext> </mml:math> produced through vector-boson fusion with a photon in the final state. These results use data from LHC proton–proton collisions at $$\sqrt{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> </mml:math> = 13 $$\text {TeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TeV</mml:mtext> </mml:math> collected with the ATLAS detector and corresponding to an integrated luminosity of 139 $$\hbox {fb}^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>fb</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . 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 $$Z\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> 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 $$1.31\pm 0.29$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.31</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.29</mml:mn> </mml:mrow> </mml:math> fb. An observed (expected) upper limit of 0.37 ( $$0.34^{+0.15}_{-0.10}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> <mml:msubsup> <mml:mn>34</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>0.10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.15</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> ) at 95% confidence level is set on the branching ratio of a 125 $$\text {GeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>GeV</mml:mtext> </mml:math> 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 ( $$0.017^{+0.007}_{-0.005}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> <mml:msubsup> <mml:mn>017</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>0.005</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.007</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> ), assuming the Standard Model production cross-section for a 125 $$\text {GeV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>GeV</mml:mtext> </mml:math> Higgs boson.