I think that it was taken at Tobolsk -- see the shape of the fence behind the Heir -- it matches with the fence's shape at Governor's mansion at Tobolsk.
I think it was taken at the same time as this:
I tried to find any photo of the similar gutter seen on the roof on the photo, but I had bad luck, though all Tobolsk's houses had the similar gutter as the one in the photo. Back in 2006, when I first saw the photo on AP, cropped, it was labeled to have been taken in February 1918 at Tobolsk. Also, the nuns visited the Governor's mansion during the IF's captivity.
(Notice the shape of the gutter -- top right and fence's shape, bottom right).
I have not any evidence to prove that it was taken at Stavka -- Trough winter 1915 Aleksei went to visit front line, he wore different outfit (mostly his uniform and cap, not winter coat, or winter coat combined with cap) and after severe nose-bleeding in December in 1915, he went back to Tsarskoe, to be back at Mogilev in May of 1916, after he recovered. During late 1916, the Heir spent his time mostly indoors, studying. This one was taken in very late December of 1916, before leaving Stavka:
