  The process $e^+e^-\to\eta\pi^0\gamma$ is studied in the center-of-mass
energy range 1.05-2.00 GeV using data with an integrated luminosity of 94.5
pb$^{-1}$ collected by the SND detector at the VEPP-2000 $e^+e^-$ collider. The
$e^+e^-\to\eta\pi^0\gamma$ cross section is measured for the first time. It is
shown that the dominant mechanism of this reaction is the transition through
the $\omega\eta$ intermediate state. The measured cross section of the
subprocess $e^+e^-\to\omega\eta\to\eta\pi^0\gamma$ is consistent with previous
measurements in the $e^+e^-\to\pi^+\pi^-\pi^0\eta$ mode. It is found, with a
significance of 5.6$\sigma$, that the process $e^+e^-\to\eta\pi^0\gamma$ is not
completely described by hadronic vector-pseudoscalar intermediate states. The
cross section of this missing contribution, which can originate from radiation
processes, e. g. $e^+e^-\to a_{0}(1450)\gamma$, is measured. It is found to be
15-20 pb in the wide energy range from 1.3 to 1.9 GeV.
