The low-lying energy levels of proton-rich Cu56 have been extracted using in-beam γ-ray spectroscopy with the state-of-the-art γ-ray tracking array GRETINA in conjunction with the S800 spectrograph at the National Superconducting Cyclotron Laboratory at Michigan State University. Excited states in Cu56 serve as resonances in the Ni55(p,γ)Cu56 reaction, which is a part of the rp process in type-I x-ray bursts. To resolve existing ambiguities in the reaction Q value, a more localized isobaric multiplet mass equation (IMME) fit is used, resulting in Q=639±82 keV. We derive the first experimentally constrained thermonuclear reaction rate for Ni55(p,γ)Cu56. We find that, with this new rate, the rp process may bypass the Ni56 waiting point via the Ni55(p,γ) reaction for typical x-ray burst conditions with a branching of up to ∼40%. We also identify additional nuclear physics uncertainties that need to be addressed before drawing final conclusions about the rp-process reaction flow in the Ni56 region.