Possible Clues:

• Don't drop one's aitches like Captain 'Ook, for example
• Say “hey,” say
• Sound of an h
• Say how, for example, snake gets angry
• H, perhaps, in the role of buccaneer?
• Like “Hook”, say, pronounce the H
• What Eliza didn't do for 'enry 'iggins?
• Record the sound of Beethoven's fifth
• Letter required when appropriate?
• It's omitted by some Cockneys - like Jack Sparrow?