Once again. Suppose there is a 1 kg weight on the bathroom scales. I remove the weight, and while still carrying it, I now stand on the scales. By how much will the weighed amount increase?
So it is with the air. There is a big column of air on the bathroom scales. As soon as I step on the scales, I remove the direct effect of the air, but it remains because it is sitting on my head, as it were.
Buoyancy, as Tom rightly says, is something else. However, buoyancy would cause my weight on average to decrease at the poles, not increase, as Tom claims. I say 'on average' because the daily change in atmospheric pressure in London is far greater than the average difference between London and the poles.
My questions remain. (1) Do things weigh more at the poles and if so (2) why is that?