We report the first high-precision mass measurements of the neutron-rich nuclei 74,75Ni and the clearly identified ground state of 76Cu, along with a more precise mass-excess value of 78Cu, performed with the double Penning trap JYFLTRAP at the Ion Guide Isotope Separator On-Line (IGISOL) facility. These new results lead to a quantitative estimation of the quenching for the neutron shell gap. The impact of this shell quenching on core-collapse supernova dynamics is specifically tested using a dedicated statistical equilibrium approach that allows a variation of the mass model independent of the other microphysical inputs. We conclude that the impact of nuclear masses is strong when implemented using a fixed trajectory as in the previous studies, but the effect is substantially reduced when implemented self-consistently in the simulation.