I know the pump work and the tube is clear (when tested pumping water in the sink). When a second person is under sink and squeezes the bottle it will pump out. What needs replacing/fixing?

Thoroughly clean the dispenser, make sure the soap isn't too thick-thin it with water and reassemble.

What needs replacing/fixing?

Nothing.

The soap is thick. When you press and get one squirt of soap it takes time for the vacuum in the tubing to pull more soap up.
Press the dispenser once and get one squirt, you must then wait a few moments before you can get another. If that is unsatisfactory dilute the soap with water.

Thoroughly clean the dispenser, make sure the soap isn't too thick-thin it with water and reassemble.

You are having a common problem called soap dismorphia, it occurs when there is a lack of fluid and ammonia in the mixture. This is going to sound just a little weird but put a little bit of urine in there until you see some, not a lot but some bubbles. Once the bubbles calm down a bit, set everything back up and you are ready to go. It is needed because you have a clog and the ammonia in the urine will dissolve it while not ruining the equipment or the soap. I was a maintenance man for like 8 years and I found this method to always do the trick 100% of the time.