Revealing mistake: In the scene after the battle of Agincort, where Henry and his retainers are walking meditatively among the heaps of the fallen, if you look carefully, you will see one of the fallen with his head raised watching the procession. He then realizes that he is on camera and puts his head back down. It's near the edge of the screen so may occasionally be cropped.
Other mistake: During the Battle of Agincourt, the same shots of soldiers being killed are used several times. Most noticeably there is one poor French soldier who dies from an arrow in his back at least twice.
Answer: Harry is a nickname for those with the given name of Henry, and has been for centuries. Today's Prince Harry, for example, is actually named Henry but is called by this nickname.