How does mudskipper breathe?

Answer 1

a direct quote from wikipedia:

• The ability to breathe through their skin and the lining of their mouth (the mucosa) and throat (the pharynx). This is only possible when the mudskipper is wet, limiting mudskippers to humid habitats and requiring that they keep themselves moist. This mode of breathing, similar to that employed by amphibians, is known as cutaneous air breathing.[4] Another important adaptation that aids breathing while out of water are their enlarged gill chambers, where they retain a bubble of air. These large gill chambers close tightly when the fish is above water, keeping the gills moist, and allowing them to function. They act like a Scuba diver's cylinders, and supply oxygen for respiration also while on land.[4]

Answer 2

Mudskippers breathe through mucus membranes located in their throat and mouth, as well as through moistened skin.