In order to repair this very important function of the climate system you will need to complete this procedures of the TECHNICAL BULLETIN issued by Jaguar:
"Vehicle Fogging/Climate Control Slow To Cool
Vehicle Air Blend Door Loose
Repair Procedure
2002-04 MY X-TYPE VIN: C00001-D99632
On some 2002-04 MY X-TYPE vehicles within the above VIN range, customers may
have concerns with the vehicle interior fogging or the climate control cooling the vehicle
slowly.
The concern may be caused by the air inlet blend door being loose on the actuator
shaft.
Customer complaint of the above condition on a 2002-04 MY X-TYPE ."
Google it for download: XT 412-02 Air Blend Door Loose.pdf
When there is no problem the climate control module set the recirculation door in about few minutes after the key is in position 0 to the fully open position in order to allow fresh air into cabin when car is stationary.
To test the correct functioning of te blend door start the climate without AC because you do not have to start the engine only turn the key to position 2 and put the ventilation to max speed.
Select the recirculation button(red led iluminating) and you should observe a change in the ventilation flow and noise too while the door closes and reach the closed point (the flow and noise will increase).
Press again recirculation button(red led not iluminating) and the flow and noise will decrease.The blend door moves to fully open position.
If you do not observe this changes the blend door does not functioning and failed.
In LHD and RHD cars you will first dismount the wipers and remove the pollen filter in order to see that the door is not obstructed in some ways.
The repair procedure is very costly if you will go at dealer or a workshop.
It is indicated to do it your self.
You have to follow this middle age repair procedure recommended by Jaguar wich practically consist of adding a drop of superglue to the motor shaft that operates the air inlet blend door.
Also, yuo will need an unlimited pacience and a lot of time with breath breaks,smoke breaks,beverage breaks and fine hands and fingers too.