If we examine the text carefully in Act IV scene 1, we can see that actually, although Henry borrows the cloak from Erpingham, it is Pistol who comes across Henry rather than the other way round. Henry, at the beginning of this scene dismisses his nobles who wait upon him with the following words:
I and my bosom must debate awhile,
And then I would no other company.
Clearly his request to borrow Erpingham's cloak and therefore his intention of disguising himself as just a normal soldier the night before a momentous battle where his forces are vastly outnumbered shows a desire to get to know what the spirit of his men is like. Let us remember that up until this stage, Henry has not truly been with his men to know how they are finding the war. This night, partly because of his troubled conscience and the way that he is commanding his men in all probability to die for him, he chooses to get to know them in an intimate way as they really are, and for this he needs to be alone and disguised.
Let us remember one way of viewing this play is that it is a play debating the nature of the perfect king. If we think of this for a moment, we can see that the perfect king must know his men and those who follow him, both the nobles and the common men. Henry, in his raucous days as Prince Hal in Henry IV Part 1 and 2, clearly knew them, and coldly rejected Falstaff and others when he became king. However, here perhaps he seeks to reconnect with the comman man before giving his famous speech so he can know how best to encourage them.