One side of my cat’s face is swollen with a hard lump that’s growing, and today I found exposed skin with slight bleeding.

Hello. From your description, your cat may have a gingival fistula. This usually comes from dental problems; if tartar, gingivitis, or periodontitis aren’t treated, the disease worsens and slowly damages the teeth, rotting the roots and causing the fistula.

For now, take the cat to a vet promptly to confirm the cause and apply the right treatment.