As Graeme said above, completely strip the carb into complete-individual pieces and clean thoroughly with carb cleaner, in a clean environment. No point cleaning a carb with dirty hands and putting the parts on a dirty table. I do in the bath room over the sink (with the plug in so I don't loose any jets down the drain!) Each clean part goes onto layers of kitchen roll.
We've had the same problem, and we got a whole new carb gasket kit as well to ensure they were not holding or letting in any dirt.
It would also be worth while washing the whole air intake setup too with some soapy water, so the the intake elbow, air filter cage, surrounding foam and filter cover.
Only when we did all of that, did we stop having problems.
So don't waste your money on a pattern part, just get the gasket kit below - or even get 2 or 3 (refurb both carbs / keep a spare gasket kit):
http://www.ebay.co.uk/itm/Kart-Genuine-Honda-GX160-Carb-Gasket-Set-Best-Price-on-eBay-/221084541716?pt=UK_Cars_Parts_Vehicles_Other_Vehicle_Parts_Accessories_ET&hash=item3379aa6314#ht_1305wt_850
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