Psoriasis affecting the outer ear and ear canal can be particularly bothersome and it can be very difficult to adequately control symptoms. The most common treatment when the ear canal is affected, is the use of topical steroid preparations in either liquid or cream form. Periodic debridement of epithelial scaling or irrigations of the ear canal may also be necessary. In regards to the itching in the tonsil region, this is most likely unrelated to psoriasis. A more likely diagnosis would be allergy or chronic tonsillitis, with chronic tonsillitis being more likely.
Chronic tonsillitis is generally a condition of adults.
The tonsils have crevaces which are called crypts. The crypts may accumulate debris, initially particles of partially digested food. This debris may coagulate and can occasionally even form small crystals or stones. Often the presence of such debris triggers inflammation which may be manifest by symptoms of mild but chronic sore throat or even as the itching you describe. Vigorous oral hygiene measures are the standard recommendation for treatment and include the use of water-pik irrigations as you have described.