We leverage the Born-Oppenheimer approximation to present a general description of topological defects dynamics in p-atic materials on curved surfaces. Focusing on the case of an active nematic, we find that activity induces a geometric contribution to the motility of the + 1 / 2 defect. Moreover, in the case of a cone, the simplest example of a geometry with curvature singularity, we find that the motility depends on the deficit angle of the cone and changes sign when the deficit angle is greater than π, leading to the change in active behavior from contractile (extensile) to extensile (contractile) behavior. Using our analytical framework, we then identify for positively charged defects the basin of attraction to the cone apex and present closed-form predictions for defect trajectories near the apex. The analytical results are quantitatively corroborated against full numerical simulations, with excellent agreement when the capture radius is small compared to the cone size.