We report the mass measurement of $^{56}$Cu, using the LEBIT 9.4T Penning trap mass spectrometer at the National Superconducting Cyclotron Laboratory at Michigan State University. The mass of $^{56}$Cu is critical for constraining the reaction rates of the $^{55}$Ni(p,$gamma$)$^{56}$Cu(p,$gamma$)$^{57}$Zn($beta^+$)$^{57}$Cu bypass around the $^{56}$Ni waiting point. Previous recommended mass excess values have disagreed by several hundred keV. Our new value, ME=$-38 626.7(6.4)$ keV, is a factor of 30 more precise than the suggested value from the 2012 atomic mass evaluation [Chin. Phys. C {bf{36}}, 1603 (2012)], and more than a factor of 12 more precise than values calculated using local mass extrapolations, while agreeing with the newest 2016 atomic mass evaluation value [Chin. Phys. C {bf{41}}, 030003 (2017)]. The new experimental average was used to calculate the astrophysical $^{55}$Ni(p,$gamma$) and $^{57}$Zn($gamma$,p) reaction rates and perform reaction network calculations of the rp-process. These show that the rp-process flow redirects around the $^{56}$Ni waiting point through the $^{55}$Ni(p,$gamma$) route, allowing it to proceed to higher masses more quickly and resulting in a reduction in ashes around this waiting point and an enhancement to higher-mass ashes.
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