  Recently it has been shown that Bern-Carrasco-Johansson (BCJ) numerators of
colour-kinematic duality for tree-level scattering amplitudes in Yang-Mills
theory (coupled with scalars) can be determined using a quasi-shuffle Hopf
algebra. In this paper we consider the same theory, but with higher-derivative
corrections of the forms $\alpha' F^3$ and $\alpha'^2 F^4$, where $F$ is the
field strength. In the heavy mass limit of the scalars, we show that the BCJ
numerators of these higher-derivative theories are governed by the same Hopf
algebra. In particular, the kinematic algebraic structure is unaltered and the
derivative corrections only arise when mapping the abstract algebraic
generators to physical BCJ numerators. The underlying kinematic Hopf algebra
enables us to obtain a compact expression for the BCJ numerators of any number
of gluons and two heavy scalars for amplitudes with higher-derivative
operators. The pure gluon BCJ numerators can also be obtained from our results
by a simple factorisation limit where the massive particles decouple.
