  All positive helicity four-point gluon-graviton amplitudes in
Einstein-Yang-Mills theory coupled to a dilaton and axion field are computed at
the leading one-loop order using colour-kinematics duality. In particular, all
relevant contributions in the gravitational and gauge coupling are established.
This extends a previous generalized unitarity based computation beyond the
leading terms in the gravitational coupling $\kappa$. The resulting purely
rational expressions take very compact forms. The previously seen vanishing of
the single-graviton-three-gluon amplitude at leading order in $\kappa$ is seen
to be lifted at order $\kappa^{3}$.
