Reasons
At first, she tells herself that she pushes him because it's the right thing to do. All of her arguments are so perfectly logical, as rational as the light of day, because he is wrong and has lost his way. She clings to her own righteousness even as she seethes. It's only in the quiet solitude of the night, when relentless dreams refuse to let her sleep, that she dares to speculate that perhaps her motives aren't so pure. The messy knot of emotions which Uther summons up deep in her gut are hard to ignore.
She longs for him to see the error of his choices and to admit for just once that he is wrong. For him to see that she is right and strong in her convictions. That she is no longer a child, but an adult to be respected. To be considered. Morgana wants him to see her; a young woman.
She tells herself that it's because she wants to be seen as an adult, as equal, that she continues to push him. He is her guardian and her King, but before that he is first a man. Intelligent and strong, but also flawed and so incredibly misguided that it makes her want to scream. She has seen the man behind the King that so few else ever got to see. Sat at dinner at the end of the day, she has seen the weariness he tries to hide behind the rim of his goblet. She has probably seen more of the real man than even Arthur has ever known. And he frustrates her at every turn.
She is young and beautiful and witty, she knows the full arsenal which she has at her disposal. And Uther will play along, will be charmed by her wit and forever compliment her for her beauty. Not once, not ever, will he see her as anything more. And so Morgana pushes him more and more, with a single minded stubbornness matched only by Uther's own.
She wants to find the point where surely he must give, where he will concede. No longer does she tell herself it's because she is right. She does it because she is not wrong and she wants Uther to see. Morgana wants him to see her.
