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

On a structure of the one-loop divergences in $4D, {\cal N}=2$ supersymmetric sigma-model