  Different types of core-collapse supernovae (SNe) have been considered as
candidate sources of high-energy cosmic neutrinos. Stripped-envelope SNe,
including energetic events like hypernovae and super-luminous SNe, are of
particular interest. They may harbor relativistic jets, which are capable of
explaining the diversity among gamma-ray bursts (GRBs), low-luminosity GRBs,
ultra-long GRBs, and broadline Type Ib/c SNe. Using the six-year IceCube data
on high-energy starting events (HESEs), we perform an unbinned maximum
likelihood analysis to search for spatial and temporal coincidences with 222
samples of SNe Ib/c. We find that the present data are consistent with the
background only hypothesis, by which we place new upper constraints on the
isotropic-equivalent energy of cosmic rays, ${\mathcal E}_{\rm
cr}\lesssim{10}^{52}~{\rm erg}$, in the limit that all SNe are accompanied by
on-axis jets. Our results demonstrate that not only upgoing muon neutrinos but
also HESE data enable us to constrain the potential contribution of these SNe
to the diffuse neutrino flux observed in IceCube. We also discuss implications
for the next-generation neutrino detectors such as IceCube-Gen2.
