A search is presented for new phenomena in events with large missing transverse momentum and a Higgs boson decaying to two photons. The search uses 3.2 fb$^{-1}$ of proton-proton collision data collected at the centre-of-mass energy of 13 TeV with the ATLAS detector at the CERN Large Hadron Collider in 2015. The results are interpreted in two theoretical models: decays of a heavy scalar into a Higgs boson and a pair of dark matter candidates, and a vector mediator emitting a Higgs boson and decaying into two dark matter candidates. No excess over the background expectation is observed and upper limits are set on the rates of these processes.

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