It's not unusual for white poodles to have those red stains. None of the four white poodles I have had over the last 40 years have had this problem, but I've known several that did. My grooming book says this is caused by oxidation of the tears. It stands to reason that if eye boogers aren't removed, they build up, dry, and irritate, causing tearing, causing oxidation, causing the red color.
When I was young, I admired clever people. Now that I am old, I admire kind people. -Abraham Joshua Heschel, theology professor (1907-1972)