Can the charged Higgs boson be discovered in $e\gamma$ collisions using the channel $e^-\gamma\to\nu H^-$?
We consider the single production of charged Higgs bosons $H^\pm$ of the two-Higgs-doublet model in $e\gamma$ collisions through the production subprocess $e^-\gamma\to\nu H^-$. As the production rate is governed by the size of the loop-induced $H^\pm W^\mp\gamma$ coupling, the cross section can only be substantial for smaller values of $\tan\beta$, $\tan\beta\lesssim1$, where the top-bottom loop contribution is enhanced. In this case, however, the natural width of the charged Higgs boson becomes large so that the Standard Model continuum background becomes important. We study the background subprocess $e^-\gamma\to\nu\bar t b$ including the interference with the signal, and find that in the region of charged Higgs mass that can be interesting, the signal is difficult to detect.