Many times Morgana tried to kill Arthur and Merlin, and each time she failed. Why didn't she simply reveal that Merlin was a warlock in order to fracture the long-term friendship between Arthur and Merlin? This would solve everything: it would separate them so they could be picked off one by one.
Morgana learns of Merlin being the mage only in the 3rd to last episode (Season 5, Episode 11, The Drawing of the Dark). Mordred tells her at the end of that episode who Emrys is (aka Merlin).
Morgana and Arthur never meet up again after that (or rather, when they meet, Arthur has learned already that Merlin uses magic).
It is because Morgana underestimate two things: Merlin himself and his relationship with Arthur. She is born and raised a noble woman. Morgana as shown previously on the serie that she does not really consider the servants on the long term. With Gwen for example: one day she wants to save her, the next kill her, and then she doesn't care anymore. Uther shows the same type of behaviour toward Gaius: one day he is a precious help, one day he should be burned, the next he is an ally, another he is ready for retirement...
It is not because Morgana does not know that Merlin has magic. She does as early as season 4. In S04E06 "A servant of two masters", she tells Melin:
I haven't seen you since you condemned my sister to a slow and painful death.
So she does know that Morgause's magical wound was cause by Merlin. He even ask her to "tell to Arthur" because he admits in front of her that:
[He] is quite proud of that achievement.
At that point, Morgana may not meet Arthur himself, but she has an influence on the court via Agravain. They are even framing Gaius for treason, which he hasn't done. They could have expose Merlin for magic, but they don't even think about it.
1x03 was about merlin and gwen, that was the secret she was going to keep.
About the others times with morgause, lets just say that morgause believe it that the one who attack her in the end of season 3 was gaius, and not only one but twice (she did not saw that it was merlin) but morgana saw merlin in there as well as gaius so that's what he's refering to in that scene (4x06)