Details of the explosion mechanism of core-collapse supernovae (CCSNe) are not yet fully understood. There is now an increasing number of successful examples of reproducing explosions in the first-principles simulations, which have shown a slow increase of explosion energy. However, it was recently pointed out that the growth rates of the explosion energy of these simulations are insufficient to produce enough $^{56}$Ni mass to account for observations. We refer to this issue as the `nickel mass problem (Ni problem, hereafter) in this paper. The neutrino-driven wind is suggested as one of the most promising candidates for the solution to the Ni problem in previous literature, but a multi-dimensional simulation for this is computationally too expensive to allow long-term investigations. In this paper, we first built a consistent model of the neutrino-driven wind with an accretion flow onto a protoneutron star (PNS), by connecting a steady-state solution of the neutrino-driven wind and a phenomenological mass accretion model. Comparing the results of our model with the results of first-principles simulations, we find that the total ejectable amount of the neutrino-driven wind is roughly determined within $sim$ 1 sec from the onset of the explosion and the supplementable amount at a late phase ($t_e gtrsim 1$ sec) remains $M_mathrm{ej} lesssim 0.01M_odot$ at most. Our conclusion is that it is difficult to solve the Ni problem, by continuous injection of $^{56}$Ni by the neutrino-driven wind. We suggest that the total amount of synthesized $^{56}$Ni can be estimated robustly if simulations are followed up to $sim 2$ seconds.
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