We study, by Monte Carlo simulations, a coarse-grained model of a water monolayer between hydrophobic walls at partial hydration, with a wall-to-wall distance of about 0.5 nrn. We analyze how the diffusion constant parallel to the walls, D-parallel to, changes and correlates to the phase diagram of the system. We find a locus of D-parallel to maxima and a locus of D-parallel to minima along isotherms, with lines of constant D-parallel to resembling the melting line of bulk water. The two loci of D-parallel to extrema envelope the line of temperatures of density maxima at constant P. We show how these loci are related to the anomalous volume behavior due to the hydrogen bonds. At much lower T, confined water becomes subdiffusive, and we discuss how this behavior is a consequence of the increased correlations among water molecules when the hydrogen bond network develops. Within the subdiffusive region, although translations are largely hampered, we observe that the hydrogen bond network can equilibrate, and its rearrangement is responsible for the appearance of density minima along isobars. We clarify that the minima are not necessarily related to the saturation of the hydrogen bond network.