  We study the fermionic Matsubara Green functions in medium-mass nuclei at
finite temperature. The single-fermion Dyson equation with the dynamical kernel
of the particle-vibration-coupling (PVC) origin is formulated and solved in the
basis of Dirac spinors, which minimize the grand canonical potential with the
meson-nucleon covariant energy density functional. The PVC correlations beyond
mean field are taken into account in the leading approximation for the
energy-dependent self-energy, and the full solution of the finite-temperature
Dyson equation is obtained for the fermionic propagators. Within this approach,
we investigate the fragmentation of the single-particle states and its
evolution with temperature for the nuclear systems $^{56,68}$Ni and $^{56}$Fe
relevant for the core-collapse supernova. The energy-dependent, or dynamical,
nucleon effective mass is extracted from the PVC self-energy at various
temperatures.
