what you have attached to the end of the cable is a plunger. not uncommon for them to get way stuck in the carb body and can be a real pain to remove. basically its a brass fitting(soft)slotted on the top and having a round spot for the cable to slip in and out, as well as a spring in between.
carb spray usually wont cut it, i would start with some sort of high quality penetrating oil like a pb blast or equivalent. patience will be your saving grace. if you have access to some quality carb soak, dripping some in there with a syringe or eye dropper may help. most important thing to remember is to save the carb at all costs, sacrifice the plunger or the cable, or both, but dont hurt the carb.
its a game of soak and wait and hope for the best. dont be afraid to try several different cleaners, oils, gunk cutters etc and dont be in a hurry or an animal with it. sometimes a well placed and tight fitting screwdriver can help if inserted in the slot on top. after the cleaner or whatever you use has had a change to work a bit, the screwdriver theory can be applied to rotate the plunger and then you can wait some more while things soak in and hopefully loosen up.
last ditch effort after exhausting all other options is to carefully drill the plunger enough to weaken it to a point to remove it. again, dont hurt that carb.