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

We study the quantum structure of four-dimensional $\mathcal{N}=2$ 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 $\mathcal{N}=2$ 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.