  The resonance scattering of Ly$\alpha$ photons with neutral hydrogen atoms in
the intergalactic medium not only couples the spin temperature to the kinetic
temperature but also leads to a heating of the gas. We investigate the impact
of this heating on the average brightness temperature of the 21-cm signal from
the Cosmic Dawn in the context of the claimed detection by the EDGES low-band
experiment. We model the evolution of the global signal taking into account the
Ly$\alpha$ coupling and heating and a cooling which can be stronger than the
Hubble cooling. Using the claimed detection of a strong absorption signal at
$z\approx 17$ as a constraint, we find that a strong Ly$\alpha$ background is
ruled out. Instead the results favour a weak Ly$\alpha$ background combined
with an excess cooling mechanism which is substantially stronger than
previously considered. We also show that the cooling mechanism driven by the
interaction between millicharged baryons and dark matter particles no longer
provides a viable explanation for the EDGES result when Ly$\alpha$ heating is
taken into account.
