A pus-y discharge from the eye needs to be treated with a local application of something such as Orbenin, not a systemic antibiotic. 'Just in case' doses of AntiBs can contribute to the development of resistance. Don't be scared to use AntiBs where they are genuinely needed, but it's not best practice to give a shot of this or that just in case it works.
I suggest you mention it to your vet, maybe take the animal to the surgery (cheaper than a home visit), and the vet can prescribe the correct product, having seen the eyes, which we have not.