the cylinder is a "push" stick. it pushes the bottom of the airless pump back down so you can refill it.
If you mean the skinny one that looks like a clear straw, it is to fill the pump with your product of choice such as liquid foundation, shampoo, etc. - it requires the tube as the 'airless' pump doesn't just open and you pour in the product...
Hard to explain, but it is for filling the pump bottle.
Hope that helps!