  Gravitational wave observations of black hole-neutron star binaries,
particularly those where the black hole has a lower mass compared to other
observed systems, have the potential to place strong constraints on
modifications to general relativity that arise at small curvature length
scales. Here we study the dynamics of black hole-neutron star mergers in
shift-symmetric Einstein-scalar-Gauss-Bonnet gravity, a representative example
of such a theory, by numerically evolving the full equations of motion. We
consider quasi-circular binaries with different mass-ratios that are consistent
with recent gravitational wave observations, including cases with and without
tidal disruption of the star, and quantify the impact of varying the coupling
controlling deviations from general relativity on the gravitational wave signal
and scalar radiation. We find that the main effect on the late inspiral is the
accelerated frequency evolution compared to general relativity, and that--even
considering Gauss-Bonnet coupling values approaching those where the theory
breaks down--the impact on the merger gravitational wave signal is mild,
predominately manifesting as a small change in the amplitude of the ringdown.
We compare our results to current post-Newtonian calculations and find
consistency throughout the inspiral.
