I don't believe we have a direct translation of that, at least not in the way I remember it used when I was in Ukraine. We just call it by its name, i.e. 'hat', 'cap', 'hood', 'headscarf', etc. BTW, nobody wears bonnets anymore.Originally Posted by gRomoZeka
Yes, you could use "headgear", "headwear", or "headdress" (for tribal natives), but it would sound weird to use those terms in everyday speech. At the very least it would sound like you were trying to humorous.
If you need to be general, I would just say 'hat', which basically covers any non-specific type of head covering.
For example:
"Я никуда не иду без головного убора."
"I don't go anywhere without a hat."