Does damaging the brachial plexus mean permanent damage or will it help eventually?

Depends on injury: The cervical plexis can be stretched as part of the critical seconds during delivery.After the head is out, it is pulled forward & angled up or down to get the shoulders & rest of baby out.If the plexis responds with swelling around the nerves, transient loss of function(dys/wks) may be found.If the nerves are damaged by the stretch, nerves may be permanently damaged.The recovery potential varies.

Not necessarely: It's pretty common to have a transient brachial plexus injury during difficult vaginal delivery of the large infants. In most cases it's temporarily from the nerve stretching. Usually time will tell. Ot/pt intervention would be also helpful, ask your pediatrician for referral.