  A nonlocal covariant extension of the two-flavor Nambu and Jona-Lasinio (NJL)
model is constructed, with built-in constraints from the running coupling of
QCD at high-momentum and instanton physics at low-momentum scales. Chiral
low-energy theorems and basic current algebra relations involving pion
properties are shown to be reproduced. The momentum-dependent dynamical quark
mass derived from this approach is in agreement with results from
Dyson-Schwinger equations and lattice QCD. At finite temperature, inclusion of
the Polyakov loop and its gauge invariant coupling to quarks reproduces the
dynamical entanglement of the chiral and deconfinement crossover transitions as
in the (local) PNJL model, but now without the requirement of introducing an
artificial momentum cutoff. Steps beyond the mean-field approximation are made
including mesonic correlations through quark-antiquark ring summations. Various
quantities of interest (pressure, energy density, speed of sound etc.) are
calculated and discussed in comparison with lattice QCD thermodynamics at zero
chemical potential. The extension to finite quark chemical potential and the
phase diagram in the $(T,\mu)$-plane are also discussed.
