not exactly sure.. but if you are reacting to strange behaviour of the temp gauge, make sure you know this first.
THe gauge does not display what the temp sensor is reading..
the gauge signal comes from the engine computer, and it is a massaged signal.

It is programmed to stay pointed dead centre as long as the temp is normal.
if your gauge is acting strangely then most likely you are having out of normal temperatures.

Thermostat is most likely issue on X-Type.

I am only guessing about your post, but I was thrown by my gauge, I suspected a temp sensor problem, but it turned out to be the thermostat.

 

There are several FUSES to check but I do not remember which ones, maybe 3,5,9. These will have to do with modules and the fans. My suggestion is to get a copy of the JTIS. It has loads of procedures to TEST and diagnose your cooling system.

In the diagram below one black arrow to FUEL sensor and the other is the Water Temp sensor.

 

I do not know if ECU cares how hot or cold the engine is running to mix air/fuel. It would really be a complicated program of comparing sensor data and then tabulate the air/fuel ratio. What I do know is that the O2 sensors are responsible for telling the ECU that the vehicle needs more air or fuel. The O2's in conjunction with the CATs make it run lean or rich. The danger of a bad O2 is that your are in danger of saturating the CATs and they are very costly. O2 are not cheap but reasonable as far as Jag parts prices go. Also if your CAT becomes saturated it will foul and damage your O2 sensor over time and you will be right back where you are now.

 

Most common reason for lean code on X-type is unmetered air getting into the intake manifold... cracked hose or orings leaking.

 

Hi, I use this parts diagram (attached) may be this can help you find any other part you may be looking for in future