Handstand trouble due to past injury (beginner handstand)

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Given such an injury, it is unsurprising that you might have a fear of falling while in a handstand -- there are many people who also have the same fear, and that's without having previously injured themselves. Being comfortable with falling, and knowing how to bail out properly, should go a long way towards lessening the fear. It also makes good sense.

If you do start to fall from a handstand towards your back, then yes, you should certainly tuck your chin in to your chest to protect your neck. Before this, though, it would be worthwhile practising simpler drills, such as making yourself feel comfortable doing simple forward rolls on padded ground from a sitting/squatted position (if you aren't already comfortable with these, that is).