This isn't strictly Clint/Natasha, but their relationship is implied. Hope that's all right.

On This Winter’s Night (6025 words) by athena4lynn
Chapters: 5/5
Fandom: Marvel Cinematic Universe, The Avengers (2012)
Rating: General Audiences
Warning: No Archive Warnings Apply
Characters: Clint Barton, Natasha Romanov, Phil Coulson
Summary:

Five Times Clint Barton Found Something He Needed In A Church On Christmas Eve