The Goshawk will have been using pretty much the same technique it'd use with a hare but as someone pointed out, finding a lethal spot would have been tricky with a mink due to it being able to squirm about in its own skin. Mink fight as a matter of course in competition with other mink and have adapted to take superficial injuries by having very thick, loose skin to stop exactly the area the Goshawk would naturally go for being vulnerable. It's exactly the same as the loose scruff at the back of a dog's neck. If grabbed and shaken by this area the skin is the area damaged, not the spinal cord.
The Goshawk is probably confused why the prey is wriggling so effectively and is not dying easily via repeated pecks and rips to the back of the neck. It's really lucky it had such a good grip with its talons or it might have become lunch itself. It was also in the classic "got a tiger by the tail" conundrum.
It's not like the Goshawk will be having thoughts about it being a mink, it's the right size, on the ground and unaware, therefore it's potential prey. Instinct takes over... Goshawks are birds, not people, and don't think in the same way as we do. It might now imprint the shape and jizz of a mink on its brain as not worth the bother or it might have another go and become a Darwinian selection if it gets eaten next time. It's highly unlikely Goshawks would specialise as a mink-eater because it's just straight up too dangerous. This was probably a mistake by an inexperienced bird.
