It might be in a pedo way? I am not sure.
But that last line I thought was more of a guardianship thing. They're talking about getting rid of Nina (as in killing her) because an institution that they're suspicious of might have knowingly sent her in order to interfere with the saintess or the kingdom in some way.
By sending her to the doctor, that might spare her from punishment if it does come to light that Nina's presence poses a threat to the King's affairs. As she is a child too, if they had to get rid of her without killing her, that still means she's homeless and possibly with a reputation for being a spy. I think it's a guardianship transfer in this context and not a giving away a person thing.