The scene where Hal and Falstaff take turns pretending to be the king to amuse themselves and the patrons of the tavern is one of the most famous in the play. It doesn't demand that the actor playing Hal tries to sound like the actor playing his father, but it is arguably funnier if he does. Tom Hiddleston, who is known to enjoy making of impressions of fellow actors, revealed in an interview that ahead of filming, Jeremy Irons had recorded himself doing the speech that Hal makes pretending to be his father in the tavern, to help Hiddleston practice his impression of the king.
The character Doll Tearsheet (played by Maxine Peake), is featured in the Hollow Crown's Henry IV, Part 1, but in the actual plays, she doesn't turn up until Part 2, where Hal and Poins hear about her and apparently don't know who she is. When she is seen in bed with Falstaff at the beginning of the Hollow Crown's Part 1, she mutters "Hang yourself, you muddy conger", which is an actual line of hers from Part 2.
When the sheriff comes to the tavern looking for those responsible for the Gadshill robbery, everybody flees the room but Prince Hal and Doll Tearsheet, so as to give the sheriff the impression he is interrupting an intimate moment and make it easier to send him away. In the actual play, the stage direction is simply for everyone but Hal and Peto to leave the room, but the choice of having Hal go at it with a woman at this point was also made by Orson Welles in his film based on the plays, Chimes at Midnight (1965). In the corresponding scene in Gus van Sant's My Own Private Idaho (1991), partly inspired by the plays, the Hal character is in bed with River Phoenix's character.