I'm really not sure what he is referring to by 出向く. As based on my understanding of the word, it wouldn't make sense for him to have gone to her (which I don't think is what it means in this context)
He asked her to do some work, that he was originally meant to do but he lacked the time to do it himself.
When she had completed the work successfully she was called in by the senate, who did not approve of him delegating his responsibility to her (he was also present, but they asked her questions about what happened).
The queen then asks her what she would like her reward to be, after that is settled she is told she can leave. He then chases after her and starts talking.
What does 出向く mean in this context? I think it could mean that he is saying that he should have been the one to stand in-front of the senate and explain things to them and she shouldn't have needed to be there, but I'm not sure.