Oops I forgot to mention treatment.
My dog got anothers dog toenail in the corner of her eye once during a spat when a friend brought their dog into our home. it went near my dogs food. I put a stop to animal visitors after that Anyways, it use to get infected and run with pus . I used polysporin especially made for eyes. It would clear it right up. But I never did run that treatment by my vet. I would always try and wash her eye out first with just a warm wet cloth. Then i would put a dab on ointment on my finer and gob it in the corner and try and wipe it inside under the lid.
I hope someone else can help you as this was my own personal treatment and not sure if the vet would have agreed with it.