Basic Purpose – When the engine is cold the choke valve is fully closed (some may have a small gap). This allows the engine to get that extra fuel it needs to get going. When the engine starts vacuum is applied to the choke pull-off which will retract, opening the choke valve slightly. How much, if any depends on the vehicle, but the bottom line is that the pull-off keeps the choke from closing too much when the vacuum drops down (engine speeds up).
Testing with a vacuum pump – Apply 15″ of vacuum to the vacuum line going to the choke pull-off. The pull-off should retract all the way and for at least 20-30 seconds. If the plunger isn’t retracted completely, replace the pull-off.
Testing with the engine running – Your engine should have at least 15″ of vacuum. This would be at idle. Again, if the plunger isn’t retracting all of the way, then replace the pull-off.
On a Quadrajet the choke pull-off has another job. At wide open throttle the pull-off controls the opening rate of the secondary air valves.
So if you are getting what seems to be a too rich condition when 1st starting your engine then the problem could be your choke pull-off, assuming your carburetor has one.