  We study the subgap spectrum of the interacting single-level quantum dot
coupled between two superconducting reservoirs, forming the Josephson-type
circuit, and additionally hybridized with a metallic normal lead. This system
allows for the phase-tunable interplay between the correlation effects and the
proximity-induced electron pairing resulting in the singlet-doublet (0-$\pi$)
crossover and the phase-dependent Kondo effect. We investigate the spectral
function, induced local pairing, Josephson supercurrent, and Andreev
conductance in a wide range of system parameters by the numerically exact
Numerical Renormalization Group and Quantum Monte Carlo calculations along with
perturbative treatments in terms of the Coulomb repulsion and the hybridization
term. Our results address especially the correlation effects reflected in
dependencies of various quantities on the local Coulomb interaction strength as
well as on the coupling to the normal lead. We quantitatively establish the
phase-dependent Kondo temperature $\log T_{K}(\phi)\propto\cos^{2}(\phi/2)$ and
show that it can be read off from the half-width of the zero-bias enhancement
in the Andreev conductance in the doublet phase, which can be experimentally
measured by the tunneling spectroscopy.
