  Based on the Dyson-Schwinger equation, we compute the resummed gluon
propagator in a holonomous plasma that is described by introducing a constant
background field for the vector potential $A_{0}$. Due to the transversality of
the holonomous Hard-Thermal-Loop in gluon self-energy, the resummed propagator
has a similar Lorentz structure as that in the perturbative Quark-Gluon Plasma
where the holonomy vanishes. As for the color structures, since diagonal gluons
are mixed in the over-complete double line basis, only the propagators for
off-diagonal gluons can be obtained unambiguously. On the other hand,
multiplied by a projection operator, the propagators for diagonal gluons, which
exhibit a highly non-trivial dependence on the background field, are uniquely
determined after summing over the color indices. As an application of these
results, we consider the Debye screening effect on the in-medium binding of
quarkonium states by analyzing the static limit of the resummed gluon
propagator. In general, introducing non-zero holonomy merely amounts to
modifications on the perturbative screening mass $m_D$ and the resulting
heavy-quark potential, which remains the standard Debye screened form, is
always deeper than the screened potential in the perturbative Quark-Gluon
Plasma. Therefore, a weaker screening, thus a more tightly bounded quarkonium
state can be expected in a holonomous plasma. In addition, both the diagonal
and off-diagonal gluons become distinguishable by their modified screening
masses ${\cal M}_D$ and the temperature dependence of the ratio ${\cal M}_D/T$
shows a very similar behavior as that found in lattice simulations.
