  We develop a systematic approach to construct the one-loop ${\cal N}=4$ SYM
effective action depending on both ${\cal N}=2$ vector multiplet and
hypermultiplet background fields. Beginning with the formulation of ${\cal
N}=4$ SYM theory in terms of ${\cal N}=2$ harmonic superfields, we construct
the one-loop effective action using the covariant ${\cal N}=2$ harmonic
supergraphs and calculate it in ${\cal N}=2$ harmonic superfield form for
constant Abelian strength $F_{mn}$ and corresponding constant hypermultiplet
fields. The hypermultiplet-dependent effective action is derived and given by
integral over the analytic subspace of harmonic superspace. We show that each
term in the Schwinger-De Witt expansion of the low-energy effective action is
written as integral over full ${\cal N}=2$ superspace.
