The interactions of water and oxygen with well-defined Cu films on the oxygen-terminated ZnO(0001)-O surface has been studied, mainly using temperature-programmed desorption (TPD) and low-energy He+ ion scattering (LEIS). The Cu films, which were from submonolayer to multilayer in coverage, had been structurally characterized in previous studies using XPS, LEIS, ARXPS, LEED, work function measurements and CO adsorption. Submonolayer films, deposited at low temperatures in the form of 2D islands, adsorb water and oxygen much like the Cu(110) surface. Upon annealing to high temperatures, these films show a shift in the Cu-derived water TPD peak from 168 down to 158 K, consistent with a conversion of the Cu film from one offering Cu(110)-like sites to one offering Cu(111)-like sites. Even heavy doses of Cu show a water TPD peak due to free ZnO(0001)-O sites, consistent with a film structure where thick Cu islands only cover most, but not all of the surface, with clean ZnO patches between. The fraction of the ZnO surface which is exposed (as measured by this water TPD peak's area) increases with the pre-annealing temperature of the Cu film above 550 K, due to 3D clustering of the Cu. Between 130 K and 370 K, the influence of submonolayer Cu islands on the water desorption temperature from these free ZnO sites decreases, suggesting that small Cu 2D islands coalesce into larger 2D islands. Oxygen-dosed, submonolayer Cu films showed no clear evidence for a hydroxyl state in water TPD.