Hi,
I have an F3 model ZX6R and had almost the same issue last year, I started by giving the bike a good service, oil, plugs, fuel filter, oil filter, air filter (K+N). This made an improvement but still didn't get rid of the spot between 6k - 8k RPM, I decided that as it was popping badly through the exhaust I would check the valve clearances.
When I stripped the top end down I found that on virtually all valves no clearances existed. I set up all the clearances in the end, it is certainly not a job for the inexperienced home mechanics and that made the bike roar once more. All I had now was a slight flatness throughout the rev range.
I rectified this the other week by balancing my cards, this was very easy and I would recommend all ZX6R owners do this at least once a year, the difference was amazing.
In light of the above I would recommend the following in this order:
Full service
Check and adjust valve clearances (If not been done in past 2 years 8K miles)
Carb balance
= What seems like a new bike
Have fun!
If you have any questions about the stuff I've done to my bike just ask!