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An integrable system is introduced, which is a generalization of the$\mathfrak{sl}(2)$ quantum affine Gaudin model. Among other things, theHamiltonians are constructed and their spectrum is calculated within theODE/IQFT approach. The model fits within the framework of Yang-Baxterintegrability. This opens a way for the systematic quantization of a largeclass of integrable non-linear sigma models. There may also be some interest interms of Condensed Matter applications, as the theory can be thought of as amultiparametric generalization of the Kondo model.

ODE/IQFT correspondence for the generalized affine $\mathfrak{ sl}(2)$ Gaudin model