Jaime intends on keeping his oath to go north to help defeat the army of the dead. This is made clear with his conversation with Cersei before he leaves:
J: You pledged our forces to fight our common enemy.
C: I'll say whatever I need to say to ensure the survival of our house.
You expect me to trust the man who murdered our father? You expect me to command our troops to fight beside foreign scum, to fight for the Dragon Queen?
J: You saw it with your own eyes.
You saw a dead man trying to kill us.
C: I saw it burn.
If dragons can't stop them, if Dothraki and Unsullied and Northmen can't stop them, how will our armies make a difference?
J: This isn't about noble houses, this is about the living and the dead!
C: And I intend to stay amongst the living.
Let the Stark boy and his new queen defend the North.
We stay here where we've always been.
J: I made a promise.
As for why he covers his hand, well he's trying to stay incognito. He's also lost his fancy armour and I also can't see his sword easily either. He doesn't want to be recognised and a big gold hand is a giveaway. See the scene here:
As @GhotiandChips says in the comments when he covers his hand it is also symbolic of him leaving Cersei. As we know she spent ages creating the hand and getting it just right even though it didn't seem useful to Jaime. See that scene here: