I read the IF in her note-- and I'd really want to see your email that you sent to her to understand it in context, as it's really not possible to get too much from just one side-- as her not being completely sure what she might have done that was wrong. She thinks it might be x and y, but she's not sure. I see her as wanting to apologize for the right thing, but she's not secure in understanding what the specific behavior might have been.

But if you don't feel that she's competent, which your words clearly indicate that is the case, and they have what I think is a demeaning edge about them, your T will feel this in session. As at least one person has pointed out, she is giving you valuable feedback. She consistently feels like she is doing something wrong, but isn't clear what it is. Which also ties right into her apology. And a t having a "lack of competency" is such a global assessment that indicates a level of black and white thinking that may permeate all your interactions with her. The other person typically experiences this as just general negativity but with little direction for how to change in the future to make things more positive.

The eggshells language is what people use to describe living with a partner who is emotionally explosive in unpredictable ways. It suggests that she feels big reactions from you at times that she doesn't understand how the reaction is tied to what's going on in session. It very well could be that she's incompetent-- I don't know, you're in the best position to judge that. But it could also be that you are harsh and reactive to not getting what you want in session, and aren't communicating well what is going on with you and what she can do differently to help you.