  We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states
in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop.
In the limit where charging energy is larger than the superconducting gap, we
determine the subgap excitation spectrum, the corresponding supercurrent, and
the differential conductance as measured by a normal-metal tunnel probe. In
absence of a phase difference only one linear combination of the superconductor
lead electrons couples to the spin, which gives a single YSR state. With finite
phase difference, however, it is effectively a two-channel scattering problem
and therefore an additional state emerges from the gap edge. The energy of the
phase-dependent YSR states depend on the gate voltage and one state can cross
zero energy twice inside the valley with odd occupancy. These crossings are
shifted by the phase difference towards the charge degeneracy points,
corresponding to larger exchange couplings. Moreover, the zero-energy crossings
give rise to resonant peaks in the differential conductance with magnitude
$4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin
do not alter qualitatively any of the results.
