Hmmm. The area where eggs and waste descend is called the vent. It's normal for it to be warm and moving in and out. But, the discharge isn't. I'm now leaning more towards an infection. I would post your last post in the https://www.backyardchickens.com/f/10/emergencies-diseases-injuries-and-cures. You may get more responses there. I would get her on some antibiotics, organic, unpasteurized apple cider vinegar and maybe some garlic.