A horse that raises its head and braces its neck when asked to back is using those body parts to resist the backward pressure — the raised head and tight neck allow the horse to pull forward against the halter pressure rather than yielding to it, much like a person leaning backward to resist being pulled forward. Fixing this requires teaching the horse to lower and soften through the neck before or during the backup rather than stiffening against it.
The first approach is to ask for lateral flexion before asking for backup — bend the horse's neck gently to one side, wait for a soft release through the poll and jaw, then immediately apply the backup cue while the horse's neck is still soft from the flexion. Many horses that stiffen their necks going straight back will back more softly immediately after lateral flexion because the neck is already in a softer, more yielded state.
The second approach is to use rhythmic pressure rather than steady pressure for the backup cue. A steady backward pull gives the horse something to brace against — it can simply set its neck and lean against the constant tension. Rhythmic pulse pressure is harder to brace against because it changes constantly, and many horses will drop their head and soften through the neck as they search for the release from the changing pressure.
The third approach addresses the root cause directly by working on poll yielding as a separate exercise — teaching the horse to lower its head and release through the poll in response to downward lead rope pressure — and then requiring that soft poll position before asking for any backup. A horse that has a confirmed poll-yield habit will be less likely to raise its head in resistance to backup pressure because the habit of lowering is stronger than the habit of raising.