  We analyse the one-loop effective action of N=4 SYM theory in the framework
of the background field formalism in N=2 harmonic superspace. For the case of
on-shell background N=2 vector multiplet we prove that the effective action is
free of harmonic singularities. When the lowest N=1 superspace component of the
N=2 vector multiplet is switched off, the effective action of N=4 SYM theory is
shown to coincide with that obtained by Grisaru et al on the base of the N=1
background field method. We compute the leading non-holomorphic corrections to
the N=4 SU(2) SYM effective action.
