Depends on how you define “self awareness” but knowing that it doesn't know something instead of hallucinating a plausible-but-wrong is already self awareness of some kind. And it's both highly valuable and beyond current tech's capability.
https://openai.com/index/introducing-simpleqa/
especially this section Using SimpleQA to measure the calibration of large language models
I'm wondering wether it would count, if one would extend it with an external program, that gives it feedback during inference (by another prompt) about the correctness of it's output.
I guess it wouldn't, because these RAG tools kind of do that and i heard no one calling those self aware.
If you have an external program, then by defining it's not self-awareness ;). Also, it's not about correctness per se, but about the model's ability to assess its own knowledge (making a mistake because the model was exposed to mistakes in the training data is fine, hallucinating isn't).
That is definitely an ability that current LLMs lack.