Hi Frankallen,
I wait for being snow free, and left with only light frost, and I unprotect them.
I removed the trashcans and the dirt I had mounded around mine. So in your case, I would remove the hay to let the sun heat the dirt and avoid rodent nesting.
Don't cut anything before July. By then, you'll know what is still alive and cut accordingly .
The choice is yours of course, but leaving the hay could disturb the lower stems and buds while trying to leaf out.