  We reconsider the prompt photon photoproduction at HERA in the framework of
the $k_T$-factorization QCD approach. The proposed method is based on the
${\cal O}(\alpha^2\alpha_s)$ amplitudes for $\gamma q\to\gamma gq$ and $\gamma
g^*\to\gamma q\bar q$ partonic subprocesses. Additionally, we take into account
the ${\cal O}(\alpha^2\alpha_s^2)$ box contributions $\gamma g\to\gamma g$ to
the production cross sections. The unintegrated (or transverse momentum
dependent) parton densities in the proton are determined using
Kimber-Martin-Ryskin (KMR) prescription. Our consideration covers both
inclusive and jet associated prompt photon photoproduction rates. We find that
our numerical predictions agree well with the recent data taken by H1 and ZEUS
collaborations at HERA. We demonstrate that the box contributions are sizeable
and amount up to $\sim15%$ of the calculated total cross section.
