'Putting on airs' is pretending to be more important/significant than you really are. I've never met 'put on Harry', though 'Harry' appears in some other idioms. My father-in-law used to complain about tools being 'blunt as Old Harry' - though maybe this was a purely Dorset saying, referring to Old Harry Rocks - Wikipedia, the free encyclopedia. In other sayings, 'Old Harry' is a euphemism for Satan.
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