Definitely looks like a puncture. Should be cleaned and an antibiotic ointment applied like neosporin. If it gets swollen it should be lanced and antibiotics given orally which usually requires a vet or someone with experience and supplies.
I have a feral cat in my home about 12 weeks old. She has a very infected eye. I tried to put drops in it and did that all wrong. It's been 5 days and she's more comfortable if I can get another outside cat to come in. But I'm giving her tiny pieces of xanax in her food when I can manage to...
It may be too late. She's in my kitchen but want to be friendly and she's crying but I cant' get near her to treat it. Was able to get ointment on shut eye. Very infected. Have antibiotic drops but she kicked them right out of my hand. It looks bad and painful. I can't afford a vet.