Molting chicken with bloody skin help
The ingredient pramoxine is pramocaine, so I would not use it. Other things than can be used for feather-picking are Nustock cream, a pine oil and sulfur cream that hides red skin and heals, Blu-Kote which colors wounds blue and contains alcohol and the anti-fungal gentian violet, Vicks, Bag Balm, or anti-peck lotions from your feed store. Make sure they are getting plenty of protein in their feed (between 16-20%) and give them a little extra protein if needed. Overcrowding can also promote feather-picking, and letting them out to free range during daylight hours really helps prevent boredom.
I vote for feather mites without specific "scientific" evidence to the contrary. A rooster or hen (depending on when it hatches) molts for the first time at about 20 months old.
These mites cause chickens to rip out their own plumage. There is nothing that will help other than a pour-on parasite treatment. The feather mites infest the Skin follicles that feathers grow out of. Nothing about a molting chicken should involve bleeding. Your hen may even be trying to get relief by pulling out her own feathers and if these feathers are regrowing they are pin feathers A.K.A. blood feathers. Damaged blood feathers however bleed like a stuck pig.