I have a vague feeling that I've also met the expression 'all Harry' to mean 'very much/extremely'. If your context refers to 'put on airs', there's a chance that 'put on Harry' means something like 'put on airs in every possible way'. But, as Whoknows has explained, 'put on' has a different meaning - although it's fairly old-fashioned. In the 1950's it was common for people to say 'You're putting me on'. Today people would say something like Whoknows's 'You're pulling my leg.' Other modern possiblities are 'You're winding me up' (/waɪ.../) ,'You're kidding me' and 'You're having me on'. I've heard these last two conflated into 'You're kidding me on', but I don't think that's a recognized idiom.
b