The origin of the bulk of the high-energy astrophysical neutrinos seen by
IceCube, with TeV--PeV energies, is unknown. If they are made in photohadronic,
i.e., proton-photon, interactions in astrophysical sources, this may manifest
as a bump-like feature in their diffuse flux, centered around a characteristic
energy. We search for evidence of this feature, allowing for variety in its
shape and size, in 7.5 years of High-Energy Starting Events (HESE) collected by
the IceCube neutrino telescope, and make forecasts using larger data samples
from upcoming neutrino telescopes. Present-day data reveals no evidence of
bump-like features, which allows us to constrain candidate populations of
photohadronic neutrino sources. Near-future forecasts show promising potential
for stringent constraints or decisive discovery of bump-like features. Our
results provide new insight into the origins of high-energy astrophysical
neutrinos, complementing those from point-source searches.Comment: 29 pages, 13 figure