Well, we are going to have to guess on this one. The image shows small purple spots with light gray centers. There are many possible fungal causes but I have note that none have been reported from the PNW. (I will check on a different data base I have access to later.) We do get powdery mildew and it does result in purple spots on the upper leaf surface but I am not confident in that diagnosis. Have a look at that description here and see if it fits:
https://pnwhandbooks.org/plantdisease/host-disease/leucothoe-powdery-mildew

Well, we have only gotten a dozen or so of these plants into the OSU Plant Clinic in the last 60+ years. There are a few of those other leaf spotting fungi which you may have. Main control tactics for these kind of problems is to remove any fallen leaves from under the plant and do not irrigate such that leaves are wet for extended periods of time. There is a home fungicide listed with powdery mildew that could help but try the non-chemical tactics first.

Thanks for the article you sent.  It does seem like this is some kind of powdery mildew.  The area where the leucothoe is planted is humid (near faucet) and has become shadier.  I’ve cleaned up the area and removed the worst leaves.

The neem oil might be helpful but you will have to get good coverage including the underside of leaves. The peroxide would not be helpful.