High temperature deposition of graphene on Cu by chemical vapor deposition can be used to produce high quality films. However, these films tend to have a non-equilibrium structure, with relatively low graphene adhesion. In this study, samples of graphene grown on copper foils by high temperature CVD were post-deposition annealed at temperatures well below the critical temperature of Cu. Resistance to etching under plasma was examined to assess the mechanical robustness of the graphene on the Cu surface, analyzed using optical and Raman microscopies. We found a correlation between the post-annealing time and etching time for the complete removal of graphene from Cu. Etching rates, minimum etch times, and surface appearance were observed to vary depending on the etching plasma (air, oxygen or nitrogen). Oxygen plasmas were found to be the least aggressive, emphasizing the improved adhesion with post-annealing treatments. Our results imply that the etching of graphene on Cu, and hence the adhesion of graphene, can be controlled by proper annealing and choice of plasma gas.