  Star-forming and starburst galaxies are considered as one of the viable
candidate sources of the high-energy cosmic neutrino background detected in
IceCube. We revisit contributions of supernova remnants (SNRs) and hypernova
remnants (HNRs) in such galaxies to the diffuse high-energy neutrino and
gamma-ray backgrounds, in light of the latest Fermi data above 50GeV. We also
take into account possible time dependent effects of the cosmic-ray (CR)
acceleration during the SNR evolution. CRs accelerated by the SNR shocks can
produce high-energy neutrinos up to $\sim100$ TeV energies, but CRs from HNRs
can extend the spectrum up to PeV energies. We show that, only if HNRs are
dominant over SNRs, the diffuse neutrino background above 100 TeV can be
explained without contradicting the gamma-ray data. However, the neutrino data
around 30 TeV remain unexplained, which might suggest a different population of
gamma-ray dark CR sources. Alternatively, we consider possible contributions of
Pop-III HNRs up to $z\lesssim10$, and show that they are not constrained by the
gamma-ray data, and thus could contribute to the diffuse high-energy
backgrounds if their explosion energy reaches ${\mathcal E}_{\rm
POP-III}\sim{\rm a~few}\times10^{53}$erg. More conservatively, our results
suggest that the explosion energy of POP-III HNRs is ${\mathcal E}_{\rm
POP-III}\lesssim7\times{10}^{53}$erg.
