If you observe your cat breathing only by its open mouth, your cat is clearly having a respiratory problem and this is a MEDICAL EMERGENCY.

Your cat may also be making sounds while trying to breathe through its mouth, but do not attempt to waste any time trying to handle this yourself. Get it right over to a vet or the nearest animal hospital and let them handle it.

Do not be tempted to stick your fingers down its throat, turn it upside down and push on its chest, or handle it in any manner other than transporting it. You could make it worse.

If your cat struggles with you, do not allow your cat to waste precious time by reacting to the stress it is obviously under. This website of the vet Greer can help you learn more.