Can a brachial plexus injury also affect your breathing?

Yes: The vast majority of cases of brachial plexus injury are not associated with breathing problems. However, in some cases the extent of nerve damage reaches higher cervical roots. If the 4th cervical root is involved, there may be damage to the phrenic nerve, which controls movement of the diaphragm. In these cases, there may be some breathing difficulty on the associated side.

Yes: When there is damage to higher brachial plexuses (involving cervical 4 part of spinal cord), it can cause see-saw breathing from the involvement of one or both sides of the diaphragm; if there is pneumothorax (air leak) because of rupture of pleura (lung cover) from the fracture of collar bone (clavicle), it can cause of severe pain and respiratory difficulty (can be life threatening), also.