I can understand why you are feeling drained OP. Living in multicultural, multiracial household takes work and whereas poc grow up continuously acknowledging, confronting and operating within our racialised society, for you having race at the forefront is new. You are getting a taste of that, it really is mentally exhausting and being honest about that is not wrong.
Gaining a better understanding of her black heritage and the experience of mixed race children is a continuous task and you will not get it right in a matter of weeks. Your expectation of yourself here is unrealistic. It’s ok that you’re struggling, as long as you’re trying. In a lot of mixed relationships the white partner not only has the poc partner but also has probably had years worth of conversations about race before children.
She is of an age where she’s navigating her identity especially being mixed so not all the thoughts she verbalises will be fully formed or acceptable. Even as an adult I feel very apprehensive when going to places with little diversity. She will probably feel a lot more comfortable once she has made some friends. If there’s anything you could do to encourage that, maybe that’s something to focus on.
The black heritage experience isn’t just racism so perhaps as well as giving her space to talk about that perhaps try and connect with her by exploring her culture through positive things. Lighthearted books by mixed or black authors, fun films with mixed or black leads etc. Then re racism if she pulls you up on anything, don’t take it as an attack on your character. I mean this in the kindest way but everyone brought up here regardless of race will have been socialised with some anti black views. Even if you do your best at some point, you might verbalise something that you don’t think is racist. Even if you think it’s ridiculous, think about it in terms of impact not intention, Google it then have a convo about it. Just showing that you will try to empathise will go a long way. Chin up OP just do your best.