I suggest cleaning the eye with warm water. I've had some ducks like this before. I'd take a warm wash cloth and gently clean it. After I'd get all the goo away I would spray VetRx mixed appropriately with water into their eyes and put 1 drop of pure VetRx down their throat. I would do this daily and within 3-4 days the infection was gone. But this is from my point of view and my own experience. Do what makes you feel comfortable! If need be, I'm always on and willing to help! You can shoot me a PM.
Also the eye or around it looks like it make have gotten scratched, and then infected. Even so VetRx can still help in cases like this. I have tons of Muscovy ducks with long claws! VetRx has always been my go to help!