What happens to the barometric height if a faulty barometer is pushed down into the mercury trough?
The height will decrease
In an accurate barometer, the space above the mercury column in the tube is a vacuum. If air enters into that space or a drop of water gets into the tube, it will immediately change into vapor in vacuum space and the air will exert pressure on mercury column due to which the barometric height decreases. The barometric height of a faulty barometer will be less than the actual atmospheric pressure.