  For the light relativistic dark matter (DM) boosted by high energy cosmic
ray, its scattering cross section with the nucleon is sensitively dependent on
the momentum-transfer and such an dependence is caused by the mediator in the
scattering. For puffy DM particle with a size, the momentum-transfer dependence
can also arise from the DM radius effect. All these momentum-transfer
dependences should be considered. In this note we study the direct detection
limits on the cosmic ray-boosted puffy DM for a simplified model with a light
mediator. For comparison, we first re-derive the direct detection limits on the
cosmic ray-boosted point-like DM. We display the limits on various planes of
parameters and find that the limits for the cosmic ray-boosted puffy DM are
stronger than for the point-like DM.
