  We study the quantum structure of four-dimensional ${\cal N}=2$ superfield
sigma-model formulated in harmonic superspace in terms of the
omega-hypermultiplet superfield $\omega$. The model is described by harmonic
superfield sigma-model metric $g_{ab}(\omega)$ and two potential-like
superfields $L^{++}_{a}(\omega)$ and $L^{(+4)}(\omega)$. In bosonic component
sector this model describes some hyper-K\"{a}hler manifold. The manifestly
${\cal N}=2$ supersymmetric covariant background-quantum splitting is
constructed and the superfield proper-time technique is developed to calculate
the one-loop effective action. The one-loop divergences of the superfield
effective action are found for arbitrary $g_{ab}(\omega), L^{++}_{a}(\omega),
L^{(+4)}(\omega)$, where some specific analogy between the algebra of covariant
derivatives in the sigma-model and the corresponding algebra in the ${\cal
N}=2$ SYM theory is used. The component structure of divergences in the bosonic
sector is discussed.
