First of all, your supervisor should not be "friends" with a client. That is crossing professional boundaries and is highly unprofessional. And she does sound rude, snappy and unreasonable.

Unfortunately, in my experience, complaining to Human Resources about a supervisor, or bringing an issue you have with your supervisor, to a Human Resources representative usually backfires.

I would just do the retraining without complaint since they never trained you to begin with, keep your distance from the supervisor, and just do everything according to the books. Basically, keep your head down and just do the work. Communicate with your supervisor only when necessary and nothing more, and do the best job you can do.

Mistakes will be made by anyone - we're all human. Chalk up the supervisor's reactions as being unreasonable and unprofessional. It's her, not you.