(I'm not familiar with the term "re homing center" but in my area we have rescues and adoption centers that are probably equivalent)
Cats who have been in group homes are often infected with the feline herpes virus. The ASPCA lists symptoms of a feline herpes infection as
- Sneezing “attacks”
- Discharge from the nose and eyes
- Conjunctivitis or pink eye (inflammation of the eyelid)
- Lesions in and around the eyes
- Eye ulcers
- Loss of appetite
There's no reason to worry, however. First, humans cannot catch feline herpes (it's a different virus than human herpes). Second, most cats recover from the symptoms of herpes.
Feline herpes is aggravated by stress, so if your new friend already had it before the re homing center, the stress of being in the center probably caused a flare up. If your cat caught it while at the center, it may take a few weeks for her immune system to control it.
However, a few cats are never able to fully rid themselves of the symptoms of feline herpes. One of my four has daily sneezing and nasal discharge from it. Otherwise he's fine and lives a normal life, we just sometimes wipe his nose (and our walls).
If your cat is unable to put the virus into remission, then you'll want to watch his eyes for worsening symptoms (sometimes eye problems from feline herpes can develop into eye ulcers). Cornell University gives the symptoms of a eye ulcer:
The clinical signs of corneal ulceration include inflammation of the tissue surrounding the cornea; seepage of discharge from the eye; clouding of the cornea; and apparent hypersensitivity to bright light. An affected cat may squint, rub its eyes, and behave as if it is having vision problems.
If you notice these signs in the future, you should take your cat back to the vet to be evaluated for an ulcer. While most ulcers will heal on their own, they can cause severe discomfort and in rare cases can permanently affect the cat's vision.
I also check with my vet regularly (once a year or so) to see if there are any new treatments for feline herpes, so that's another thing you can do.
The good news is that you already took your cat to the vet and confirmed that she doesn't have an eye ulcer. You can clean the discharge if it seems to bother her (use a damp, warm washcloth), or just leave it.
Even though I'm not a vet, feline herpes is extremely common in rescued cats, so it's a reasonable suspicion. Your vet probably sees this problem daily and didn't take the time to explain it. I don't have a particularly good feeling about a veterinarian that doesn't realize as common as it is to THEM, it's new to YOU.