The electroweak production of $Z\left(\nu \overline{\nu }\right)\gamma$ $$ Z\left(\nu \overline{\nu}\right)\gamma $$ in association with two jets is studied in a regime with a photon of high transverse momentum above 150 GeV using proton–proton collisions at a centre-of-mass energy of 13 TeV at the Large Hadron Collider. The analysis uses a data sample with an integrated luminosity of 139 fb −1 collected by the ATLAS detector during the 2015–2018 LHC data-taking period. This process is an important probe of the electroweak symmetry breaking mechanism in the Standard Model and is sensitive to quartic gauge boson couplings via vector-boson scattering. The fiducial $Z\left(\nu \overline{\nu }\right)\mathrm{\gamma jj}$ $$ Z\left(\nu \overline{\nu}\right)\gamma jj $$ cross section for electroweak production is measured to be ${0.77}_{-0.30}^{+0.34}$ $$ {0.77}_{-0.30}^{+0.34} $$ fb and is consistent with the Standard Model prediction. Evidence of electroweak $Z\left(\nu \overline{\nu }\right)\mathrm{\gamma jj}$ $$ Z\left(\nu \overline{\nu}\right)\gamma jj $$ production is found with an observed significance of 3.2σ for the background-only hypothesis, compared with an expected significance of 3.7σ. The combination of this result with the previously published ATLAS observation of electroweak $Z\left(\nu \overline{\nu }\right)\mathrm{\gamma jj}$ $$ Z\left(\nu \overline{\nu}\right)\gamma jj $$ production yields an observed (expected) signal significance of 6.3σ (6.6σ). Limits on anomalous quartic gauge boson couplings are obtained in the framework of effective field theory with dimension-8 operators.