The adsorption of atomic H and H2 on copper mono- and submonolayers supported on hexagonal WC(0001) surfaces has been investigated using density functional theory with the Perdew–Burke–Ernzerhof exchange correlation functional and D2 van der Waals corrections. Results evidence the impact of the termination of the carbide substrate on fundamental properties of Cu adatoms, and, hence, on the stability of molecular and atomic hydrogen, defining copper's catalytic activity for hydrogen evolution reaction. Using H adsorption energy as a descriptor, catalytic activity of Cu adlayers for hydrogen evolution reaction was estimated using traditional volcano curves and a curve, obtained at low hydrogen coverage. Obtained results evidence that copper adlayers supported on the WC may present a viable low-cost alternative to noble metal-based catalysts, with improved catalytic activity compared to that of copper. This, potentially, can be a useful basis for designing and developing novel functional materials with predetermined catalytic properties.