I have two tiny toy poodles and both would get the tear stained eyes. The little white one always had problems with itchy skin too. So I began about a year ago adding apple cider vinegar to their drinking water. They haven't had either of these problems since.
A word of caution, vinegar can remove the enamel from their teeth, the same as it does for people. I have always added 1/8th teaspoon of baking soda to one-teaspoon vinegar before I add it to their water. Let the vinegar/baking soda mix sit until the fizz stops then add it to the water.
I could have missed it but I didn't see anyone saying they were doing that mix. We don't want to fix one problem while causing another.