A search for a chargino$-$neutralino pair decaying via the 125 GeV Higgs boson into photons is presented. The study is based on the data collected between 2015 and 2018 with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $pp$ collisions at a centre-of-mass energy of 13 TeV. No significant excess over the expected background is observed. Upper limits at 95% confidence level for a massless $\chi ~10$ are set on several electroweakino production cross-sections and the visible cross-section for beyond the Standard Model processes. In the context of simplified supersymmetric models, 95% confidence-level limits of up to 310 GeV in $m(\chi ~1\pm /\chi ~20)$, where $m(\chi ~10)=0.5$ GeV, are set. Limits at 95% confidence level are also set on the $\chi ~1\pm \chi ~20$ cross-section in the mass plane of $m(\chi ~1\pm /\chi ~20)$ and $m(\chi ~10)$, and on scenarios with gravitino as the lightest supersymmetric particle. Upper limits at the 95% confidence-level are set on the higgsino production cross-section. Higgsino masses below 380 GeV are excluded for the case of the higgsino fully decaying into a Higgs boson and a gravitino.