It means that the 'something white and feathery' was getting in the way of the blue sky that she could see through the window. Across is just used here to show that there is movement in front of the sky :)

I just did a Google search. I had wondered if it could be "Anne of Green Gables." I was right. In future, when you quote something in italki, it helps if you give the source of the quotation. Here's what I wrote before I found the answer. "She's just woken up. Her eyes are gummy. The sun is in her eyes. She doesn't know exactly what she is seeing. We don't know exactly what she is seeing. It is probably a complicated scene sky, sun, trees, clouds, and other things. 'Glimpses of blue sky' means that something is in between her and the sky. She can see little patches of blue sky. Maybe they appear and disappear. 'Something white and feathery' could be a white, feathery cirrus cloud in the sky. It could be white puffs from a cottonwood tree. It could be a white bird, as you guessed. She doesn't know yet. We don't know yet." We get the answer a few paragraphs later: "But it was morning and, yes, it was a cherry-tree in full bloom outside of her window." Furthermore, we read that "it was so thick-set with blossoms that hardly a leaf was to be seen." "Something white and feathery" is a bough of the tree, covered with white cherry blossoms, moving in the breeze. When cherry trees come into bloom, they bear white blossoms. This web page shows a photographer's project. It shows a girl dressed like Anne, posed in front of cherry trees with white blossoms: https://tinyurl.com/y7nvl55v

i get it, anne saw the glimpses of blue sky.