Regardless of what the injection is for, as a former needle phobic myself, please don't give in to the phobia.
You won't be doing her any favours. Needles are essential for many aspects of healthcare and unavoidable if she and any future children are to stay healthy.
It was signing up for nurse training that "cured" me, not the training but the being a ruddy pin cushion for basically what felt like the first term!
I used to faint and cry and all sorts I still don't like them nobody does. But I can do it if needed, even blood draws (and I have crappy tiny blood hoarding veins!)
Emla cream helps, as has diazepam on occasion, also getting her to focus on not the injection but that she's doing something amazing for her health and that it literally will be over in seconds.
What if that type of thing works will depend what exactly the fear is of. Is it the pain? Is it the fear of fainting (that was an issue for me after one time when a nurse didn't believe me I was prone to fainting and wouldn't let me lie down, I fainted during the blood draw and I fell off the chair and hurt my shoulder!)? Is it the fear of embarrassment of her reaction? Is it the fear of the injection being harmful? Etc you and even a nurse can discuss this and address her fears with her.
My favourite phrase comes in here "this too shall pass" applies to a LOT in life! I even apply it to getting injections or blood drawn "it'll be over soon" or any other painful experience (constipation with piles following c section- yea thanks for that dd! 😂)
Sounds silly but it works - organise something nice for the 2 of you to do after, can be as small as coffee & cake or as big as a trip or holiday that's up to you. Bribery & corruption works a treat on kids & adults alike! Give her something to look forward to.
Dd went through a phase of hating needles and I had to find a way of getting her over it, her timing was shit cos she was in hospital with a serious infection at the time. I ended up asking if nurse would cannulate me so I could "show" her it wasn't that bad, that helped a bit, plus I promised her a treat when she was out of hospital. Planning the treat was a good distraction while it was done (she needed to be on a drip). Nurse was great, used a small needle to site, was chatty and cheerful but didn't faff or "dig about" when she didn't get in first time (dd inherited my crappy veins bless her) simply withdrew and tried again. Got her second time.
Unfortunately I think some nurses are put off by such patients when really in that situation they need to at least appear confident and no nonsense without being stern