Hydrogen electrode reaction (HER) was investigated for a copper coordination polymer, catena-mu-N,N'-bis(hydroxyethyl)dithiooxamidatocopper(II) (CuCP) by galvanostatic transient measurements using a palladium cathode. Individual steps in the Volmer-Tafel reaction were successfully observed for the colloidal CuCP in alkaline solution under hydrogen absorption. The Volmer step was found to be promoted with CuCP and its hydrogen-absorbed polymer (CuCPH) from shorter decay times in transient curves and smaller overpotentials in Tafel plots. In contrast, the inhibition of the Tafel step was observed, which is due to the larger overpotentials. Difference in reducibility between the polymers and plausible reduction mechanisms is discussed.