I’ve had to deal with this twice in my horse & like others say: Eyes are No Bueno, Call Vet for me.
Staining both times showed no tear, just abrasion & I had the pleasure (NOT!) of getting ointment in the eye 3X daily for a week.
My solution to treating was to put the ointment on the tip of my finger, then dab that in the corner of the eye.
I know, not sterile, but got 'er done with 16h of noncompliance VS 5’2" me.
My softish fingertip was easier for horse to accept than the pointy end of the tiny ointment tube.
Doing the Not A Big Issue Dance for your mare!