  The one-loop process e,gamma->e,h for intermediate Higgs masses is
considered. Exact cross sections for unpolarized and longitudinally polarized
beams are computed and found to be more than two orders of magnitude larger
than the rates for the crossed process e,e->gamma,h, in the energy range
\sqrt{s}=(0.5-2) TeV. We show that, apart from being competitive with the
gamma,gamma->H process for testing the one-loop gamma-gamma-H vertex, by
requiring a final electron tagged at large angle the channel e,gamma->e,h
provides an excellent way of testing the Z-gamma-H vertex.
