Final answer:
The structure most likely responsible for the cricket's loss of coordinated movements is the apodeme.
Step-by-step explanation:
The structure that is most likely responsible for the cricket's loss of coordinated movements after the removal of a thin dome-shaped cuticle near a joint in its leg is the apodeme. An apodeme is an ingrowth on the arthropod's exoskeleton to which muscles attach. Contraction of muscles attached to the apodeme enables coordinated movements. Thus, when the cuticle with the apodeme is removed, the cricket loses its ability to make coordinated movements.