A way that can help you is to train the cat to use the cat flap somewhere else than where it is installed.
When I trained my own cat, I first didn't install the cat flap. Instead, I enclosed the cat in a carrier (a situation she doesn't like much) and fitted the cat flap into the carrier opening instead of the front grille. I then placed a bowl of food in front of the flap, outside the carrier. Then I pushed through the flap a few times with my finger to have her recognize that the flap is movable. It took her about two evenings of this training to recognize the mechanism and realize that she can go through it.
Then I installed the flap in its permanent hole, and showed the cat that she can go through. When the cat would wait for me to let her out, I would instead bring her to the flap (for me, it is in a french window beside the door itself) and push the flap open for her. I don't remember if I also physically pushed her once or twice, not completely through the whole flap, but just enough that her nose lifts it and she gets haptic feedback.
Of course, not all cats are equal, mine was under a year when she learned. But it might be a good way to train yours, if the training at the installed flap doesn't work. Especially putting her in a situation where 1) she is craving to leave and 2) she is not accustomed to solving in a different way (with you opening for her) might help her try out new things like pushing the flap.
One thing which makes cats mistrust the flap is encountering it locked. If yours is lockable, make sure to always have it unlocked until she gets accustomed to it. Even then, each single encounter of a locked clap can result in the cat asking you for opening the door for a couple of days afterwards (mine does this - I think she also dislikes the flap closing on her tail in principle, as well as enjoys my attention).
If the cat learns how to use the clap on her own (and you are sure about that - I think mine at first thought that I have to touch it before she can pass, because of the demonstrations) but still prefers you to open the door, the only solution is to outstubborn her. Mine knows very well when I am pointedly ignoring her, and if she asks me for door service and I don't give it, she at some point gives in and goes through the flap.