The game is out of sync with cloud servers follow these steps to fix it.
*This should work with challenges and letters*
Firstly, I copied the Battlefield V Profile Save Data file to USB Storage Device and also uploaded a copy to Online Storage.
Then I went back into the PS4 Settings, Application Saved Data Management, Delete, Battlefield V and deleted the BFV file.
After deleting the file, I restarted BFV, replayed the introductions, then went into Play, War Stories, Nordlys, Acts, Still and Silent (yours might be different depending on which objective bugged out for you), and replayed the mission only focusing on completing the missing objective Destroy all searchlights (yours might be different).
When I completed the objective, an in-game message popped up indicating it completed. Then I went to BFV options and quit back to the game menu.
When I checked the War Story, the Destroy all searchlights objective showed as complete, but I was missing the previously completed objective info within the individual War Story Acts.
Next, I exited BFV completely, went back into the PS4 Settings, Application Saved Data Management, Saved Data on USB Storage Device, Copy to System Storage, Battlefield V, selected Battlefield V Profile Saved Data and Copy.
This returned the previously saved data back to the PS4 and all of the objectives I'd already completed as well as the previously bugged one was now showing correctly as completed.
Again, if you try this make sure you make USB backups before you attempt this work around.
thanks to u/Troop_66 on reddit for solution.