  We provide irreducible expressions for the metric variations of the
gravitational action terms constructed from the 17 curvature invariants of
order six in derivatives of the metric tensor i.e. from the geometrical terms
appearing in the diagonal heat-kernel or Gilkey-DeWitt coefficient $a_3$. We
then express, for a four dimensional spacetime, the approximated stress-energy
tensor constructed from the renormalized DeWitt-Schwinger effective action
associated with a massive scalar field. We also construct, for higher
dimensional spacetimes, the infinite counterterms of order six in derivatives
of the metric tensor appearing in the left hand side of Einstein equations as
well as the contribution associated with the cubic Lovelock gravitational
action. In an appendix, we provide a list of geometrical relations we have used
and which are more generally helpful for calculations in two-loop quantum
gravity in a four dimensional background or for calculations in one-loop
quantum gravity in higher dimensional background. We also obtain the
approximated stress-energy tensors associated with a massive spinor field and a
massive vector field propagating in a four dimensional background.
