This note presents searches for the invisible or partially invisible decay of a Higgs boson with a mass of 125 GeV produced in collisions through vector boson fusion with a photon in the final state. This search uses data from LHC collisions at = 13 TeV collected with the ATLAS detector corresponding to an integrated luminosity of 139~fb. The event signature is characterized by a significant amount of unbalanced transverse momentum and a photon in the final state, in addition to a pair of jets with a large pseudorapidity difference. The data are compared to a combination of simulated and data-driven predictions of the signals and expected background processes, and an observed (expected) upper limit of 0.37 (0.34) at 95% confidence level is set on the branching ratio of a Higgs boson with a mass of 125 GeV to invisible particles, assuming the Standard Model Higgs boson production cross-section. The search for a signature combining the reconstructed photon and the unbalanced transverse momentum also probes the decay of the Higgs-like boson to a photon and a dark-photon. An observed (expected) 95% confidence level upper limit on the branching ratio for this decay is set at 0.014 (0.017), assuming a mass of 125 GeV for the Standard Model Higgs boson production cross-section.