Angle-integrated cross-section measurements of the 56Ni(d,n) and (d,p) stripping reactions have been performed to determine the single-particle strengths of low-lying excited states in the mirror nuclei pair 57Cu−57Ni situated adjacent to the doubly magic nucleus 56Ni. The reactions were studied in inverse kinematics utilizing a beam of radioactive 56Ni ions in conjunction with the GRETINA γ-array. Spectroscopic factors are compared with new shell-model calculations using a full pf model space with the GPFX1A Hamiltonian for the isospin-conserving strong interaction plus Coulomb and charge-dependent Hamiltonians. These results were used to set new constraints on the 56Ni(p,γ)57Cu reaction rate for explosive burning conditions in x-ray bursts, where 56Ni represents a key waiting point in the astrophysical rp-process.