The wings and nest are those of the Augurey, a creature with which one of the main characters (Delphini Diggory) feels a great affinity, even to the point of using the name as her alter ego and having a tattoo of its wings on the back of her neck.
ALBUS: Thank you. Oh, nice mark.
[DELPHI’s cloak has loosened. An Augurey tattoo is visible on the back
of her neck.]
ALBUS: On your back. I hadn’t noticed it before. The wings. Is that what the Muggles call a tattoo?
DELPHI: Oh. Yes. Well, it’s an Augurey.
SCORPIUS: An Augurey?
DELPHI: Haven’t you met them in Care of Magical Creatures? They’re sinister-looking black birds that cry when rain’s coming.
Wizards used to believe that the Augurey’s cry foretold death. When I
was growing up my guardian kept one in a cage.
The Augurey is greenish black. It is intensely shy, nests in bramble
and thorn, eats large insects and fairies, flies only in heavy rain,
and otherwise remains hidden in its tear-shaped nest. The Augurey has
a distinctive low and throbbing cry, which was once believed to
Fantastic Beasts and Where to Find Them
The "intriguing winged nest" itself contains a young boy, presumably Harry's son Albus (based on the ginger hair and forlorn expression). The boy is trapped in the nest of the Augurey, as Albus is trapped in Delphini's machinations throughout much the play. The play is also about Albus trying desperately to leave the metaphorical nest of his famous family.
The overall image is also clearly intended to look like a Golden Snitch.