My cat sneezes and the discharge is thick and a bit red, like yellow mucus with blood

A cat sneezing with nasal bleeding may be from capillary rupture during a violent sneeze, but first consider a respiratory infection, a common cold. Also consider viral diseases like feline rhinotracheitis or calicivirus.

If the cat is newly from a breeder or pet shop, get a PCR test at the vet and treat by result. If a visit is impractical, use antibiotics such as oral doxycycline with eye and nose drops.