A search for Dark Matter (DM) is presented, exploiting final states containing a photon and missing transverse momentum, using 139 fb$^{-1}$ of data collected by the ATLAS experiment (LHC) at a centre-of-mass energy of 13 TeV. The discriminant variable is the missing transverse momentum (\met): several signal regions are defined in bins of \met\ to enhance the analysis sensitivity, and associated control regions (CRs) are built to derive background expectations by normalizing Monte Carlo (MC) yields to data through a likelihood fit simultaneously performed over all the CRs. This strategy allows combining inputs from all the CRs coherently, taking into account the correlations of systematic uncertainties among different regions. No deviations are observed relative to the predictions of the Standard Model and 95% confidence level (CL) upper limits between 2.4 fb and 0.5 fb are set on the visible cross section of physics beyond the Standard Model, in different ranges of \met. The results are interpreted as exclusion limits in simplified DM models, where weakly interacting DM candidates are pair-produced via an s-channel axial-vector or vector mediator. The search excludes mediator masses below 920-1470 GeV for DM candidate masses below 280-580 GeV at 95% CL, depending on the couplings. These results are translated into exclusion limits on the interaction cross-section between DM and nucleons as a function of the DM mass, and compared with the limits currently set by direct detection experiments, underlying a good complementarity between the two detection strategies. In addition, the results are expressed in terms of 95% CL limits on the parameters of a model with an axion-like particle produced in association with a photon.