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If no hand animation appears above the mob, run through these in order:

Are you crouching? The pat requires the crouch key to be held at the moment you right-click. Release it too early and the pat will not fire.
Are you close enough? You need to be within normal interaction range. Step right up to the mob.
Is your crosshair on the mob? If you are aiming at a block, fence, or the ground, the click goes there instead.
Is it actually a living mob? Armor stands, boats, and item frames cannot be patted.

Check your keybinds. If crouch or right-click has been rebound, use your actual bound keys. A common cause is a mod or resource pack that changes default controls.
Try an empty hand. Some mobs react to held items before the pat check runs. An empty hand removes that conflict.
Test on a simple mob. Try a cow or pig in an open area with nothing else nearby. If that works, the issue is environmental — usually a block or another entity in the way.
Reload the world. Occasionally an entity's state can get stuck; leaving and re-entering the world refreshes it.

Work through these one at a time and you will almost always find the cause.