Looks like a House Finch nest (I think) with two cowbird eggs. I wouldn't expect much issue with moving the wreath slightly... just try to do it sometime when the adult(s) aren't right there.
Personally I would take out the cowbird eggs. House Finch feed their young seeds and plant material, which cowbird young can't survive on, and almost always starve. They might kill/push out some of the finch eggs or young before they do, and chances are there won't be any surviving young in the nest at all.