He allows the hanging of Bardolph, a long-term friend, acting as a lesson for others that stealing will not be permitted and even the acquaintances of the King are in danger if they do it. This does help to show his benevolence because as Pistol enters, with a favour to beg of Fluellen: Pistol's good friend and fellow-soldier, Bardolph, has been found guilty of stealing from the conquered French town. He has stolen a "pax" -- a tablet made out of some valuable material and used in religious rites. If Henry were scheming he may have allowed the theft of the religious object, but in an act of respect for the French town this action must be punished. Henry has already previously warned that "We would have all such offenders so cut off," meaning that he hopes all looters will be hanged, without exception.
This presentation may make him a bit of both: hypocrite and genius. When Hal turns against Falstaff it is truly upsetting, a terrible scene, and you see that Hal (now Henry) is a sort of monster. He has ignored Falstaff while he is ill, and the King is blamed for Falstaff's situation. Henry's "bad humours" towards him are to blame. But he is also the most successful king in Shakespeare's History plays. You can also argue that Hal has a strategy, his wildness turned majesty is the only possible tactic by which he could assert control. Hal again illustrates this when he insists, "When lenity [gentleness] and cruelty play for a kingdom, the gentler gamester is the sooner winner". Still, we might wonder at this statement when we remember that King Henry is using it to justify hanging a man -- an old friend of his, moreover, whom he now pretends not to recognize. He is not exactly being gentle by hanging a fellow soldier, not least a friend of his, and this might also weaken his army and he might have lost the battle, by this he is being hypocritical. If you were to contrast between these words and his warning to the Governor of Harfleur it is striking; telling a city ruler that he must surrender or "look to see.
