To say that Morgana's first weeks in Camelot were stressful, upsetting, and confusing, would be an understatement. She is used to freedom, to affection, and very, very few rules. As long as she stayed on her father's estate, did not purposefully go anywhere perilous (the river that ran through the estate had rapids, and her father was always afraid she would drown), and attended her lessons, Morgana's free time was hers to do with as she pleased.
In short, she finds Camelot very restrictive. She has a maid to help her dress in the morning, as if she is incapable of doing it herself. She can feel the eyes of guards (though she has little respect for their ability to actually keep an eye on her) when she is wandering outside. It's this constant supervision, not understanding that as the ward it puts her in harm's way almost as much as Prince Arthur, that she finds stifling. In the space of a little under two months, she's lost her father, her home, her nurse, most of her possessions, and her freedom. She arrived in Camelot with her horse, some of her clothes (the King insisted that he could provide new ones, so all she was allowed to bring enough until the new ones are ready), a few trinkets from her room, and the sword her father had given her for her ninth birthday. That is a lot to lose for anyone, never mind a 10 year old girl who now finds herself with only her title to her name. The estate had been absorbed into the crown's coffers, though she will not know what that means for several years.
Thus, if anyone in the palace had consistent parenting skills, they could have predicted that the very first opportunity Morgana has to slip away, she would. Having learnt of both the secret passageway out to the river, and where to find the girl, Gwen, Morgana's recently squelched independence is making a fierce counteroffensive.
She is dressed in her warmest dress -- one she brought from home, not one that has recently appeared in her wardrobe (they are too fancy to play in, why can no one see that?) -- and her fur cloak. On her hands are rough cloth mittens (also from home -- a farmer's wife made them for her when Morgana paid a visit to the woman's daughter, who had been laid up with a broken leg and unable to play for several weeks) and she carries a little bit of food in a pouch tied around her waist.
The snow in Camelot has been heavy this winter, but she thinks she can make the journey in a little over an hour, but, before she realizes it, several have passed, as she's been getting stuck in snow well above her knees. Still, she's not lost, as she's following the river, but when she takes a step, she does not realize it's into a snowdrift built up against the steep edge running along the bank of the river. One second she's following the river, and the next, she's scrabbling backwards, trying to keep her feet away from the thin ice and the open water.