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Here is a link that might be useful: mower running

IS it doing that under load?

Try loosening the gas cap for a few seconds. Does it run "normally" again for a period of time? (vent)

You might also try a strong dose of carb cleaner IN the fuel.
I'll do that once a year in the Spring.
Start with less gas in the tank then what it takes to mow.
Add about 10X a "normal" amount.
Mow till you run out.
Refill with fresh gas and finish mowing to "purge" the fuel line etc. of the concentrated dose.
You don't want to leave the high concentration in the tank for an extended time.

This post was edited by bill_kapaun on Tue, Sep 17, 13 at 11:14

Here's a fast check for adequate fuel flow; with engine running, push the top of the governor arm to the rear and hold it for two seconds. If the engine is getting adequate fuel, it will accelerate quickly and run strong. Without adequate fuel, it will hesitate and run rough. The governor arm lever is the long vertical lever at the LH rear side of the engine block. The Honda engine has an internal governor which operates this arm to pull the throttle open when engine speed drops. With your engine running and 'hunting', you will see this lever moving with every surge of the engine.
The question is, why is the engine surging? Is the governor causing it or trying to correct it?

I can try loosening the gas cap. what exactly does under load mean. So what carb cleaner would you recommend? So your suggesting putting a little gas and a lot of cleaner and run that then filling with fresh gas?

Govenor? The mower has a choke and the lever to run the mower. Sorry I am not very knowledgeable.

Most any "generic" carb cleaner from the auto parts store.

Read the label for the recommended dose.
Use about 10X that , but with only enough fuel in the tank so that you run out shortly before getting finished mowing.
Refill with fresh gas and finish mowing so that all carb cleaner has been flushed out of the system.

You DON'T want the concentrated solution sitting in the tank/lines/carb for an extended time.

The governor limits the top speed of the engine to about 3200RPM in order to prevent over-speeding the blade tips. With the throttle on 'fast', that's the speed it will run if everything is in normal condition. You 'bypass' the governor to rev up the engine momentarily for testing by manually moving the governor arm.
Also, old or moisture-contaminated fuel can contribute to this kind of problem, but with your use of stabilizer I doubt that's the case here.