the sore area is probably where the "tail" of the nail is still embedded under the skin at the corners, and where the old nail has now separated from the skin in the middle - almost like a cut as you say.
if you're feeling brave, you could start to slowly peel the old nail off from what's growing underneath - just take it very easy and it will slowly start coming off. If parts remain firmly attached, just cut the loose areas off and wait a bit longer before having another go. I've just done that with a big toenail - much more comfy now!
be prepared to be greeted with a very ugly partly grown new nail - the exposed nailbed will be sensitive to pressure for a couple of days but soon settles.
can you tell I've lost a few nails over the years?? lose one, grow one, lose that again, grow another one and so on.