Here's what happened. The stick was already lying across one of the eggs, but not obvious:

Part of it was across the bar as Kay landed. (By the way, notice how Kay just squeezes her wing past the pole the left. You can see her right wingtip there.):

When Kay lands, she jostles the stick, and it moves between the two eggs:

She spots the stick over the eggs and steps into the nest bowl to move it:

She picks up the stick and places it at the edge of the nest bowl:

Video to follow shortly.
"Freedom of the press is guaranteed only to those who own one" –– A. J. Liebling