The Venturi
(red cone that the air hose feeds into in your 3rd picture ) is either plugged or the pump is not pulling in enough water to make the skimmer work properly.
The lack of a bubble storm inside the skimmer is obvious in the pictures you posted
A new skimmer might have a piece of plastic flashing inside the air intake or the hole is partially blocked to prevent enough air from entering and mixing with the water to create the storm that makes a skimmer work.
Otherwise the pump is not working properly? A damaged impeller (broken blades or defective drive magnet, something that prevents smooth movement to get the water flowing through and pulling in air?
Maybe it was not reassembled properly?
Or it’s just a defective unit? Take it apart or take it to the lfs and get someone to troubleshoot it so you know what is wrong. It should be making a fraction of some foam after 4 months, even with a light load algae, food, fecal pellets and detritus would make some bubbles exit the cup if it was pulling water and air properly.
Take it to the fish shop and put it on one of their tanks and let them decide if it’s working or not.
Hth.