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Is the noise a single clunk when you turn it on or is a constant clunking noise or popping noise?
Ok. That is a door actuator motor making the noise. The Mustang has 4 mode door motors. One on the drivers side and three on the passenger side behind the glove box.
See the diagrams below.
It is possible the battery disconnect caused it. Drive it a while and listen for it to come back. If it does, replace the actuator. Then operate it back and forth by the switches for the AC mode selector. It will relearn.