I honestly don't know what exactly it is, but I'll try to help you narrow it down. Please keep in mind that I'm not a vet and the information given in this post is not supposed to replace a professional diagnosis by a vet.
Yeast infection: no.
Yeast infects the inner ear and produces lots of brown or black ear wax that smells bad. It looks a lot like ear mites (picture below).
Ear mites: no. Ear mites also infect the ear canal. They bite the skin so much that in a few days the entire ear canal is covered in a thick flaky crust of scab. The cat would scratch very very excessively and you would certainly notice. Here's a picture of what ear mites would look like:
Fleas: probably no. Those like to hide in warm places, not at the tips of the ears. You can try parting the fur of your cat at the arm pits or between the hind legs. If you see tiny black dots moving, it's most likely fleas. But this skin rash doesn't look like it's caused by fleas.
Lice: probably no. Lice are much bigger than fleas and you would see them crawling around in the fur. You could also see tiny white dots sticking to the hairs - those are the eggs of the lice.
Ringworm: maybe. Ringworm is a fungal infection and usually causes hair loss in circular pattern and flaky skin like your cats ear. But if the cat doesn't have bald patches in addition to the flaky skin, it's probably not ringworm. Other fungi seem to mostly cause swellings and skin lesions, no flaking.
Feline acne: maybe, but not on the ears. Many cats have dark lumps on the chin, lips or nose due to a plastic allergy. To avoid the symptoms, don't use any plastic bowls for food or water and avoid plastic toys. Use stainless steel, glass or porcelain bowls instead. You didn't post pictures of the lump your cat has, but feline acne typically looks like this, this or this:
Bacterial infection: very likely, but it's not the underlying problem. The flaky skin and tiny red dots look like the hair follicles might be infected by bacteria. The problem is that this is most often just a secondary symptom. Something else weakens the immune system and if you (or the vet) only concentrate on treating the bacterial infection, it will flare up again and again as long as the underlying problem isn't treated.
Skin mites: most probably. The pattern of the flaky skin looks like some parasite might move over or through the skin. Microscopic skin mites could do that, but I cannot tell which type of mites might create this particular flaky skin. There are several different parasites that cause what's commonly called scabies or mange. One specific type of mange affects primarily the ears.
Notoedres Cati is highly contagious and obnoxiously persistent. It typically starts around the ears and face before spreading throughout the body. These mites can only survive a few days off the host. If untreated, Notoedres cati can proliferate and cause emaciation, anorexia, and even death. (source)
Home diagnosis: You can try a rudimentary diagnosis at home, but you need an actual microscope or a microscope camera (at least 12x magnification, the more the better). Take a short strip of clear sticky tape, push it onto the hairless, flaky skin of the ear and pull it away again. Then try finding anything that moves under the microscope. If it looks like walking dandruff, it's almost certainly some type of mite. Please put the sticky tape into a sealable container and bring it with you to the vet to make a certain diagnose easier.
Home treatment: If you suspect your vet won't diagnose or treat the cat correctly, you can try home remedies and see if his symptoms improve.
- Either wash all the pillows, beddings, toys or other objects your cat came into contact with on a hot cycle, or throw them out. Mites can survive in textiles for several days and reinfect you and your cat.
- Over the counter mange treatments are available in pet stores or online. Please read the package leaflet and put a cone collar on your cat if the product isn't safe to ingest.
- You can try a topical application of apple cider vinegar. Please dilute the vinegar 50/50 with water to avoid irritating the skin of your cat even more. Then soak a cotton pad in the diluted vinegar and wipe the skin. Make sure not a single drop of the liquid drops into the ear canal.
- Plain natural yogurt applied to the skin can also help (because yogurt is acidic and mites hate acid). Since your cat will groom and lick the yogurt, make sure it does not contain the sweetener xylitol (it's poisonous to cats).
- Cortisone cream can reduce the itchiness and help the skin healing faster. You must make sure your cat doesn't ingest any of the cream, so please put a cone collar on him for the duration of the treatment.
These home treatments are suggested by holistapet.