A search for a chargino-neutralino pair decaying via the 125 GeV Higgs boson into photons is presented. This study is based on the full Run 2 data collected between 2015 and 2018 with the ATLAS detector at the LHC, corresponding to an integrated luminosity of 139~\mathrm{fb}^{-1} 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 $\tilde{\chi}^{0}_{1}$ are set on several electroweakino production cross sections and the visible cross section for beyond the Standard Model physics processes. In the context of simplified supersymmetric models, 95% confidence-level limits of up to 315 GeV in $m(\tilde{\chi}^{\pm}_{1}/\tilde{\chi}^{0}_{2})$, where $m(\tilde{\chi}^{0}_{1})$ is 0.5 GeV, are set. Limits at 95% confidence level are also set on the $\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{2}$ cross section in the mass plane of $m(\tilde{\chi}^{\pm}_{1}/\tilde{\chi}^{0}_{2})$ and $m(\tilde{\chi}^{0}_{1})$.

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