  Dark matter (DM) from freeze-in or superWIMP production is well known to
imprint non-cold DM signatures on cosmological observables. We derive
constraints from Lyman-$\alpha$ forest observations for both cases, basing
ourselves on a reinterpretation of the existing Lyman-$\alpha$ limits on
thermal warm DM. We exclude DM masses below 15 keV for freeze-in, in good
agreement with previous literature, and provide a generic lower mass bound for
superWIMPs that depends on the mother particle decay width. Special emphasis is
placed on the mixed scenario, where contributions from both freeze-in and
superWIMP are similarly important. In this case, the imprint on cosmological
observables can deviate significantly from thermal warm DM. Furthermore, we
provide a modified version of the Boltzmann code CLASS, analytic expressions
for the DM distributions, and fits to the DM transfer functions that account
for both mechanisms of production. Moreover, we also derive generic constraints
from $\Delta N_\mathrm{eff}$ measurements and show that they cannot compete
with those arising from Lyman-$\alpha$ observations. For illustration, we apply
the above generic limits to a coloured $t$-channel mediator DM model, in which
case contributions from both freeze-in through scatterings and decays, as well
as superWIMP production can be important. We map out the entire cosmologically
viable parameter space, cornered by bounds from Lyman-$\alpha$ observations,
the LHC, and Big Bang Nucleosynthesis.
