You don't really answer all of my questions. I suspect this is going to take some time to pry the information out.
Isn't the "J" hose the fresh air intake? IF so then the pump would work with the "J" hose broken off. The cap serves to keep moisture out of the pump.
IF the pump works, then the source of the P0411 must be from another cause. What have you done to rule out the other "possible" causes?