  Long-duration gamma-ray bursts (GRBs) have been often considered as the
natural evolution of some core-collapse supernovae (SNe). While GRBs with
relativistic jets emit an electromagnetic signal, GRBs with mildly relativistic
jets are opaque to photons and, therefore, could be detectable through
neutrinos only. We discuss the possibility that successful GRBs and mildly
relativistic jets belong to the same class of astrophysical transients with
different Lorentz factor Gamma_b and study the production of high-energy
neutrinos as a function of Gamma_b, by including both proton-photon and
proton-proton interactions. By assuming a SN-GRB connection, we find that the
diffuse neutrino emission from optically thick jets with Lorentz factors lower
than the ones of successful GRBs can be one of the main components of the
observed IceCube high-energy neutrino flux. Moreover, under the assumption that
all these jets belong to the same class of astrophysical transients, we show
that the IceCube high-energy neutrino data provide indirect constraints on the
rate of non-successful jets, favoring a local rate lower than tens of percent
of the local SN rate. These limits are currently comparable to the ones
obtained in dedicated searches on choked sources and are expected to become
tighter with accumulation of more high-energy neutrino data.
