what updates/changes were made to the new R56 base MINI engine to bump up the MPG to 28/37, 32 combined VS 25/33, 28 combined (manual transmission)? R56 is slightly bigger, heavier and makes 3 more HP, so I wonder what has been done to yield a significant improvement in fuel economy.

 

R53 (and the specs you gave) is the 1st gen S. Therefore there isn't a supercharger......that is why it gets better gas mileage

But seriously, assuming you meant to say R50 vs R56 (base), I would say the increase in gas mileage (though not substantial) is mostly because the R56 has MPFI (Multi-Port Fuel Injection) where the R50 did not. But thats my observations. Could be wrong.