  Employing the standard hard-scattering approach and the running coupling
method we calculate a class of power-suppressed corrections $\sim
1/Q^{2n},n=1,2,3,...$ to the electromagnetic $\pi^0\gamma$ transition form
factor (FF) $Q^2F_{\pi\gamma}(Q^2)$ arising from the end-point $x \to 0,1$
integration regions. In the investigations we use a hard-scattering amplitude
of the subprocess $\gamma+\gamma^{*} \to q +\bar{q}$, symmetrized under
exchange $\mu_R^2 \leftrightarrow \bar{\mu}_R^2$ important for exclusive
processes containing two external photons. In the computations the pion model
distribution amplitudes (DA's) with one and two non-asymptotic terms are
employed. The obtained predictions are compared with the CLEO data and
constraints on the DA parameters $b_2(\mu_0^2)$ and $b_4(\mu_0^2)$ at the
normalization point $\mu_0^2=1 GeV^2$ are extracted. Further restrictions on
the pion DA's are deduced from the experimental data on the electromagnetic FF
$F_{\pi}(Q^2)$.
