How do you feed your pet a wellness treat?

Make sure your sim has a wellness treat in their inventory and then use them to click on the sick pet. There should be an option that says something like "feed vet treat" that you can select, and it will bring up a menu where you can pick from any of the vet treats your sim has on them.

Ok, thanks so much! I must have tried to give it to the dog while it was sick earlier, when it had a cone on.

Yes, it will cure anything.

If you tried to give it to a dog wearing a cone, it wouldn't have worked, because the dog was no longer sick. They only get the cone after being treated at the vet clinic.

I often let my pet owning sims have a table or shelf where they store those wellness treats. I put most of them in the main sim's inventory, but by keeping this spare storage, they are also available to other household members (in case main pet person is away) + I avoid running out because I see when it's time to visit a vet to restock.

I'd prefer if we could turn off the pet sickness though. I don't mind them being sick, but the visuals do easily ruin any story telling, stealing my gaming time because I need to remove that red glowing nose or legs etc.