  Radiation damping of the motion of charged particles in relativistic,
optically thin plasmas is described within the framework of the covariant
gyrokinetic theory. It involves description of the collisionless
single-particle dynamics as well as the Vlasov and Maxwell equations both
written in the covariant formulation. The damping causes corrections to the
phase-space trajectory of the particle, as well as to the form of the kinetic
equation itself, due to the failure of conditions of the Liouville theorem.
Both effects result independent of the gyrophase, which is retained as an
ignorable variable. In addition, the applicability range of the covariant
gyrokinetic theory is extended to describe short-wavelength perturbations with
the background of zero parallel electric field. The presented theory is
suitable for description of magnetized, relativistic, collisionless plasmas in
the context of astrophysical or laboratory problems. Non-uniquenes of the
gyrokinetic representation and consequences thereof are discussed.
