I'm assuming you stuck the toothpick into the tube on the outside of the cup because it would be really difficult to get it into the end of the tube on the inside, at the bottom. If there is any of the toothpick sticking out the end, tweezers might work, but gently grabbing it with your sprue cutters would work even better. Just apply enough pressure for the blades of the cutters to just barely bite into the wood and slowly pull it out. You could even try kind of a prying action, pivoting the sprue cutter blade against the end of the cup tube. If all else fails, the law of diminishing returns comes into play, and sometimes its just going to be better to start over with a new cup.
https://www.paascheairbrush.com/H-1-4-OZ
I have had my Paasche H for more than 35 years and have made a lot of mistakes with its care, so it was nice to find out that the parts are easy to get and relatively inexpensive. In lieu of mechanical cleaning, what I do now is to drop the cup, air cap, and needle into a relish jar that I have filled about 2/3 full with MEK. Then I close the jar and put that in my ultrasonic cleaner with some water in the cleaner tub to help transmit the sound waves to the jar. Turn that on for 30 minutes, use latex gloves to grab the parts out of the jar (use latex, not vinyl...MEK will melt the vinyl), rinse them all under hot water, let them air dry, and you'll have nice, clean airbrush parts with nothing lodged in them.