I'm sorry that your question wasn't answered in a timely manner. We don't have many avian vets on this site. We don't apply products to the skin of finches lest they ingest what we apply to their detriment. Instead, we separate those finches with raw lesions away from aggressors. It's important to differentiate feather picking from self-mutilation of feathers.
Because self-mutilation is limited to areas that a bird can reach with its own beak, feather loss on the breast, tail or wings suggests that the bird is plucking its own feathers. If feathers are missing on the head, however, another finch is likely to be the culprit. Self-induced feather loss is also caused by an environment that generates stress and frustration. Try to improve the size and security of your bird's cage. Reduce the number of birds housed together and keep those that pluck feathers aways from larger finches that may intimidate them by their size. Even large, peaceful finches may make smaller birds nervous. If a self-mutilating bird is caged alone, provide a mate for it so it won't be lonely and bored.
Feather picking usually refers to unnatural feather loss cuased by one bird removing feathers from another. This commonly results in a bald spot on the bird's head, although truly zealous feather pluckers will not confine their picking to only the head. The primary cause of feather picking is frustration created by a stressful environment - often a cage that's too small or too crowded. All finches require adequate space for physical exercise. Confined to a small cage, these normally energetic birds become bored and frustrated. They also develop excess energy that they can't release without being disturbed by their companions. Without this privacy, they become stressed and irritable. If you have more than one pair of finches in an enclosure less than 30 inches long and feather plucking occurs, then overcrowding is likely to be the cause. Enlarge the enclosure or put some of the birds into other cages. Cages that are located in stress-inducing settings may also cause feather picking. Finch cages shouldn't be placed where family, friends or household pets such as cats, dogs or large parrots constantly pass by without warning. Frequent interruptions, loud noises or other disturbances make finches nervous and frustrated. To vent their frustrations they may pull feathers out of their companions. Feather-picked individuals should be separated until their feathers grow in again; otherwise, the bald spot may invite continued feather picking from other birds. If a bird is continually picked in one area, it may lose its protective plumage permanently making the bird more susceptible to illness.
Thank you to the Hamilton & District Budgerigar Society, Inc from whom the above was taken. Please respond with further questions or concerns if you wish.