Hello, I'm Dr. Kara. I have over 20 years of experience as a veterinarian and I'd like to help with your concerns about Kormi's head shaking and touching her left ear.
Cats display this type of behavior when their ears are bothering them. It sounds like she doesn't have the typical redness and discharge that points toward a routine yeast or bacterial infection but that doesn't mean that she isn't uncomfortable and there isn't something there.
Were any samples taken of the white patch in her ear?
If not I recommend taking some and looking at them under the microscope for signs of mites (demodex, sarcoptic or ear mites also known as otodectes), bacteria or yeast. I also recommend taking a sample and running a fungal culture. If these aren't definitive for a diagnosis then sometimes we need to take a biopsy of the affected tissue.
In the meantime while we are waiting for results I recommend trying antihistamines and omega 3 fatty acids to try and reduce her discomfort.
You can try either of these antihistamines:
1) Benadryl (diphenhydramine only, the combination products with decongestants and acetaminophen are toxic to cats) at 1mg per pound every 8 hours. That's a half of 25mg tablet every 8 hours for a cat that is 8 to 15 pounds.
2) Chlorpheniramine at 4mg once or twice daily.
Combined with high doses of omega-3 fatty acids antihistamines work to relieve her
itchiness. I like 3V caps or Derm Caps. Together antihistamines and omega-3 fatty acids work synergistically. These should help at least somewhat. Be aware that antihistamines can cause drowsiness or hyperactivity which should resolve with continued use.
Food allergy is another reason for itchy ears. You might wish to try a hypoallergenic food for 8 weeks or so. My suggestions are Hills Z/D or Royal Canin Duck and Green peas. She will need to eat only that food, no other treats or foods during her food trial.
After her diagnostic samples from her ears are collected her ear can be gently cleaned daily with an ear cleaner such as Oticlens. This will help make her ear inhospitable for bacterial or yeast growth while we wait for test results.
Best of luck with your girl, please let me know if you have any further questions.