The list of missions that need to be replayed on Hard or more difficult are listed below.

Air & Space Museum	American History Museum	Bank Headquarters
Capitol Building (Stronghold)	DCD Headquarters	District Union Arena (Stronghold)
Federal Emergency Bunker	Grand Washington Hotel	Jefferson Plaza
Jefferson Trade Center	Lincoln Memorial	Potomac Event Center
Roosevelt Island (Stronghold)	Space Administration HQ	ViewPoint Museum

Everything is more or less the same as when you played them on the 'Story' difficulty just a little more difficult. There will be more enemies of higher levels (Purple instead of Red, Yellow instead of Purple) but overall the difficulty isn't that much more difficult.

You can check your progress under the '' -> 'Progression' -> 'Commendations' -> 'Service' -> 'Merit Badges'
There is no 'you've done these' and 'these are the rest you need'. You have to manually go through all of the 'Merits' and see your 'Progress' of how many out of 10 times you have done a mission on hard or more. Your goal for this trophy is to get at least 1/10 on all 10 of the missions listed above. If you search by Alphabetical (Press ) you can just start at the top and go down the list checking each one off.