Those Star Trek Into Darkness deleted scenes that were removed from the DVD/Blu-ray release in order to make you buy a second better (or decent) Collector’s Edition release later are now available for your perusal. They fill more than a few of STiD’s plotholes, including why Carol Marcus is suddenly British.
https://gizmodo.com/the-star-trek-into-darkness-deleted-scenes-you-didnt-s-1535711446
Kirk and Pike’s discussion:
https://www.youtube.com/watch?v=GeYewTaXvS4
A Klingon conversation:
https://www.youtube.com/watch?v=RwIFhHOfdIk
Captain’s log:
https://www.youtube.com/watch?v=uTNjd6AMO_U
Marcus receives a call:
https://www.youtube.com/watch?v=nocOwybJoTw
Pike gets shot (extended version):
https://www.youtube.com/watch?v=cF8S5O0K8Jc
Klingon fight:
https://www.youtube.com/watch?v=7ew41unlpZE
Carol explains her accent:
https://www.youtube.com/watch?v=B1WbA8Ek6A4
Scotty sneaks onto the Killerprise:
https://www.youtube.com/watch?v=ub2mJ4jSwJo
Kirk’s meets with Mickey from Doctor Who’s family:
https://www.youtube.com/watch?v=PJaVTX6HAFo
Klingon dance party:
https://www.youtube.com/watch?v=2kSaLNhj8OA
H/t to Aki!