Have you checked your vacuum?  if you have a intermittent vacuum leak, you will get hesitation and surging. One thing I have noticed with these cars is the need for a well sealed up engine.  The PCV line and valve in the valve cover, one of the biggest culprits is the oil filler cap.  Eliminate the easy stuff by checking those simple things first.  We all overlook the obvious sometimes to get caught up in the more complicated gadgets and goodies. 

The surge/jerk/cough has happened twice more, again under the same circumstances of being under power in 4th gear, revs at about 4000-4500. It's almost a silent engine cough, but it's a pretty solid jerk. I will disconnect the tuning module (mentioned earlier in this thread) and try to recreate the symptoms. To Don's question about the tires, ironically, I just replaced my tires for that very reason; one of the four that I'd purchased in August was discovered to be 7 mm smaller in circumference than the other three, and neither the manufacturer nor the online retailer would stand behind the tire. So that's a $400 lesson learned. Anyway, this retailer knew to check that issue and so I don't believe tire circumference could cause this symptom. Any other ideas related to the performance tuning module idea?

Have you checked tires for equal wear and rolling diameter/ circumference? These might be the cause of your surging sound you originally described. The sudden jerk or clunk might be a sudden release of stress between mis-matched wheels, when one tire slips for an instant.

Today I installed the following engine upgrades:Mini Cooper Chip ECU Power Module N18 Gen2 R55 R56 R57 R58 R59 R60 R61

Mini Cooper S Ignition Upgrade Kit Gen2 R55-R61

I hoped adding the tuning chip module and upgrading the ignition coils and spark plugs would increase my horse power. The information with the tuning chip suggested there would be a 20 to 40 minute time lag before modifications from the performance module were effective. So after 30 minutes of normal driving, I finally shifted down at about 30 miles an hour to third gear and punched it. The RPMs build quickly until somewhere past 4500 there was a tremendous jerk or lurch. I backed off power and drove normally. Nothing seemed amiss. Five minutes later I tried it in fourth gear starting at 40 miles an hour. The revs build quickly but the lurch or jerk occurred again. Again no significant change afterwards To suggest anything was wrong. I drove normally until I got home. Did my clutch just slip? The new stage two clutch is on back order! There was no pop or a bang or noise of any kind. Traction control was off. Sport mode was engaged. Driving on dry asphalt, outside temperature 5°. Any ideas?

Turbo car's in general use a boost pressure sensor, that when bad can cause surging with no trouble code. Common problem on Ford Ecoboost engines, I don't know on the Mini's.

You are correct, I have the turbo.
No lights of any kind on the dash. I do suspect something amiss in the driveline, but the 4 short shafts have all been removed and balanced; the main shaft down the center has replaced. I'll be replacing the clutch and flywheel. I hope that catches it.

You said surging sensation

this was a problem at constant speed my sons 2004 BMW Mini with superchargers. A software upgrade was performed by the dealer and solved the problem. Meanwhile some other guys came up with some modification of the pressure boost or pop off valve. Again, supercharger,  you prob have turbo. 

Are those joints replaceable without replacing the entire drive shaft for that particular wheel?

I wondered about that, but the ride in curves is quite smooth... I will be more deliberate in checking at lower speeds under load

Can you sense the same thing when going around corner? Long sweeping curve? Thinking CV joints.

I have a 2012 Countryman ALL 4S, manual. I'm planning to upgrade to a stage 2 clutch, light flywheel, as well as upgraded ignition coils. However, cruising at highway speeds, there is a subtle surge, almost like two instruments are out of tune. The speed and RPM remain the same, but I can feel a subtle surge... i hesitate to say vibration. I cannot really hear anything, but it's clearly there. A related issue MAY be the vibration that occurs low in the RPM's of the first three gears, under hard acceleration (thus the intention to replace the clutch and flywheel). Without having driven the car, can anyone attest to similar feelings/experiences? I know the clutches are infamous in the countryman of this vintage. But the tires are new, and well-balanced. I even had all four drive shafts balanced, and I replaced the main center shaft. There are no leaks or issues with the transmission or transfer case that either I or my mechanics (or the dealership) can isolate. I'm open to your ideas and comments.
Mike