The purpose of the spring that sits on the outside of the link from the governor arm to the throttle butterfly, strangely enough, is to prevent what Honda calls surging and I call oscillating, but is much the same as hunting.
Unless the spring is in pretty bad shape I doubt it is causing your problem, but to me it doesn't make sense to ignore a fault in the anti-oscillation system when your governor is oscillating.
At this point I don't know whether your spring was in good condition before you broke it, so I suppose it is plausible that it could be at the root of your problem. Hence I think you need to replace the spring.
For reference, the anti-oscillation spring on the small OHV Honda engines pulls the throttle butterfly open slightly when the engine is stopped and the speed control is in the stop position. I'd say it holds the throttle at something like 10-20% open.
