In short: This will unlock after you beat the boss and is not missable.
In full: Immediately after beginning the story mode you will be told to choose a crew to challenge. You will then receive options for races. After a few races the game will tell you to choose another crew to challenge. Despite this, you have actually NOT beaten the first crew yet. You will know you have beaten a crew when you challenge the leader of that crew in a boss battle. Boss battles appear late into the game and will occur in the same order you chose to challenge each crew. After beating the boss you will own the district.
Boss battle tips:
You will go head to head with another car. Put most of your focus towards making good turns and not crashing or running into the wall. Use your boost if you find yourself behind. Using the boost while you are in the lead is somewhat pointless because the CPU cars always manage to catch up to you again, so save it. If possible, try to save some boost for the end of the race to give you a little push, making sure you win.