  Cosmic metals are widely believed to be produced by supernovae (SNe) and
compact object mergers. Here, we discuss the nucleosynthesis of
neutrino-dominated accretion flows (NDAFs) with outflows in the core-collapse
SNe (CCSNe), and show that the outflows from NDAFs can have a significant
contribution to the $^{56}$Ni abundance in the faint explosions if the masses
of the progenitor stars are within about $25-50$ $M_\odot$. Less massive
progenitor stars can produce more $^{56}$Ni than their more massive
counterparts in the NDAF outflow nucleosynthesis channel. Therefore, we find
that the total (i.e., CCSNe and NDAF outflows) $^{56}$Ni mass per CCSN depends
only weakly upon the mass of progenitor stars. In the metallicity evolution,
the ratio of $^{56}$Fe (decayed by $^{56}$Ni) mass to the initial total gas
mass can increase by $\sim$ 1.95 times if the upper limits of the
nucleosynthesis yields from NDAF outflows and CCSNe are considered. Our results
might have significant implication for chemical evolution of the the solar
neighborhood, galaxies, and active galactic nuclei.
