The β decays of very neutron-rich nuclides in the Co-Zn region were studied experimentally at the National Superconducting Cyclotron Laboratory using the NSCL β-counting station in conjunction with the neutron detector NERO. We measured the branchings for β-delayed neutron emission (Pn values) for Co74 (18±15%) and Ni75-77 (10±2.8%, 14±3.6%, and 30±24%, respectively) for the first time, and remeasured the Pn values of Cu77-79, Zn79,81, and Ga82. For Cu77-79 and for Zn81 we obtain significantly larger Pn values compared to previous work. While the new half-lives for the Ni isotopes from this experiment had been reported before, we present here in addition the first half-life measurements of Co75 (30±11 ms) and Cu80 (170-50+110 ms). Our results are compared with theoretical predictions, and their impact on various types of models for the astrophysical rapid neutron-capture process (r-process) is explored. We find that with our new data, the classical r-process model is better able to reproduce the A=78-80 abundance pattern inferred from the solar abundances. The new data also influence r-process models based on the neutrino-driven high-entropy winds in core collapse supernovae.