Hello "asturtarmac", and welcome to NSR-WORLD.COM
The 2T oil pumps are generally very reliable, and in almost 30 years of NSRs, I've personally never seen one fail on just one side. The most common "failure" of the oil pump is the drive shaft oil seal unseating from it's housing, and allowing all the 2T to drain into the transmission*, but even this isn't common. That is not to say however, that the pump cannot fail -- just that it's not common.
You probably know already, but just to be sure, the exhausts on the R5/6K "cross over", so the RHS tailpipe actually corresponds to the LHS/upper cylinder/carb.
To see one side start oiling up while the other remains "normal" would suggest a carburetion or ignition problem, rather than an oil pump problem.
I would start off by pulling the airbox, cleaning the air filter, and checking you don't have any debris partially blocking a jet. Also check the condition of the spark plugs, caps, and HT leads.
Please take the time to check the fault-finder in Chapter 1/Page 26 of our Honda Approved English Service Manual in the Members Content area of the site.
Please note, we do not allow jetting discussion (other that stock settings) in the public forums, as every scenario and condition is different. What works for someone may not work for someone else, and could have (and has in the past indeed had) catastrophic consequences. You can find the stock information here: NSR-WORLD Technical
*As a side note, I always recommend topping off the 2T oil tank before ever ride, that way if the oil pump drive shaft oil seal has unseated and alllowed all the oil to drain into the transmission, you will discover the issue before starting the motor.
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Andy.
NSR-WORLD.COM
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