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

An extended study of the prompt photon photoproduction at HERA with $k_T$-factorization