We investigate Kondo correlations in a quantum dot with normal and superconducting electrodes, where a spin bias voltage is applied across the device and the local interaction $U$ is either attractive or repulsive. When the spin current is blockaded in the large-gap regime, this nonequilibrium strongly-correlated problem maps into an equilibrium model solvable by the numerical renormalization group method. The Kondo spectra with characteristic splitting due to the nonequilibrium spin accumulation are thus obtained at high precision. It is shown that while the bias-induced decoherence of the spin Kondo effect is partially compensated by the superconductivity, the charge Kondo effect is enhanced out of equilibrium and undergoes an additional splitting by the superconducting proximity effect, yielding four Kondo peaks in the local spectral density. In the charge Kondo regime, we find a universal scaling of charge conductance in this hybrid device under different spin biases. The universal conductance as a function of the coupling to the superconducting lead is peaked at and hence directly measures the Kondo temperature. Our results are of direct relevance to recent experiments realizing negative-$U$ charge Kondo effect in hybrid oxide quantum dots [Nat. Commun. textbf{8}, 395 (2017)].
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