There is a circuit in the AFM that controls the fuel pump to a degree. With early FI systems like the Bosch L-Jetronic on the X, most manufacturers used either an oil pressure switch or contacts in the AFM to determine whether or not the engine was running as part of the safety logic.
Specifically for the X, this task is handled by a pair of contacts in the AFM. With the ig switch in Run (as opposed to Start), if the engine is running there should be air flow thru the AFM. If there is no airflow, then the safety circuit contacts cut power to the fuel pump. This prevents the fuel pump from running and keeping the fuel system pressurized in an emergency situation such as a collision, etc.
When the ig switch is in Start, OTOH, the fuel pump is always energized. The "changeover" between behavior in Start the behavior in Run is one of the tasks handled by the FI system's double relay.
So if you suspect that the fuel pump is not being properly energized, you would need to proof it for both Start condition and Run condition.
To check for proper behavior in Run condition, detach the air ducting from the air cleaner to the AFM, put the ig switch in Run position, and use for example the eraser end of a pencil inserted into the AFM from the air cleaner side to slightly displace the flapper door in the AFM. If everything is right, just the slightest movement of the flapper door should energize the pump which of course will be easily heard as you are right there with the engine off.
Checking for pump operation in Start condition is a little harder as the noise of the engine keeps you from hearing the pump. You could try feeling for operation while a helper turns the ig key to Start, or you could use a simple test light to check for current to the pump.
If your tests indicate that there is something wrong with the contacts in the AFM, the black plastic top of the AFM can be removed with some effort to inspect the circuitry inside. There have been some posts that show what the inside of the AFM looks like and which thingies are the fuel pump contacts. The one below is just what Google came up with from a Jaguar forum, but the accompanying post says the red arrow points to the fuel pump contacts.