Yes, you can remove a door lock (you have to start by taking the door apart on the inside to disconnect the link rods and remove the U-shaped metal cylinder retainer), disassemble it for cleaning, re-assemble it with graphite dry lube dust, and re-install. It's the kind of fine-detail work a locksmith does all the time, and I've done it often on my cars. Be aware you'll have to figure out a few tricks to getting it apart and together, and there will be several tiny parts and springs trying to jump around on you. Lose any and you're in deep doo-doo! So if you're good with fine mechanical things, give yourself time and a clean workspace and work through it, slowly the first time. But if not, only go as far as removing the lock cylinder system from the door. Take it to a locksmith to let them clean it, then re-install yourself.
I highly DON"T recommend pushing into the lock a stick with Super Glue on it. Where do you think the super Glue will go? Besides, you'd never get a solid enough grip on a jammed twig to be able to remove it cleanly.
