During the X-ray bursts of GS 1826$-$24, "clocked burster", the nuclear reaction flow that surges through the rapid-proton capture process path has to pass through the NiCu cycles before reaching the ZnGa cycles that moderate the further extent of hydrogen burning in the region above germanium and selenium isotopes. The $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction located in the NiCu cycles plays an important role in influencing the burst light curves as found by Cyburt et al. (2016). We deduce the $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction rate based on the experimentally determined important nuclear structure information, isobaric-multiplet-mass equation, and large-scale shell model calculations. Based on the isobaric-multiplet-mass equation, we propose a possible order of $1^+_1$ and $2^+_3$ dominant resonance states and constrain the resonance energy of the $1^+_2$ state. The latter reduces the contribution of the $1^+_2$ dominant resonance state. The new reaction rate is up to a factor of four lower than the Forstner et al. (2001) rate recommended by JINA REACLIB v2.2 at the temperature regime sensitive to clocked bursts of GS 1826$-$24. Using the simulation from the one-dimensional implicit hydrodynamic code, KEPLER, to model the thermonuclear X-ray bursts of GS 1826$-$24 clocked burster, we find that the new $^{57}$Cu(p,$\gamma$)$^{58}$Zn coupled with the latest $^{56}$Ni(p,$\gamma$)$^{57}$Cu and $^{55}$Ni(p,$\gamma$)$^{56}$Cu reaction rates redistributes the reaction flow in the NiCu cycles and strongly influences the burst ash composition, whereas the $^{59}$Cu(p,$\alpha$)$^{56}$Ni and $^{59}$Cu(p,$\gamma$)$^{60}$Zn reactions suppress the influence of the $^{57}$Cu(p,$\gamma$)$^{58}$Zn reaction and diminish the impact of nuclear reaction flow that by-passes the important $^{56}$Ni waiting point induced by the $^{55}$Ni(p,$\gamma$)$^{56}$Cu reaction on burst light curve.