I have an XK8 1999. If you turn the key once there's a loud click and the engine does not turn. Turn the key a second time - same thing. Turn the key a third time and the car starts up. This only happens in the morning. I've installed a new battery. There's no change.

 

If there is a Denso starter mounted in your car (most likely) prosumably the solenoid contacts of the starter are pretty burned.

There is no need to replace the starter. there are repair kits available, look for Denso starter solenoid repair set.

To get the upper bolt of the starter out is a bit difficult, to get it back in place is a pita.

Fritz

 

What you are describing sounds like a bad/poor connection for the power circuit.
I am assuming that the battery is at 12.5 to 13 volts when trying to start. Check the voltage and also clean polish and tighten the cable connectors. I would start there first as most Jaguar problems are voltage related.
My next check would be the bulkhead connector that runs from the battery in the trunk to the engine compartment on the right side of the engine compartment.
Next I would make sure that the connector for the power to the starter is clean and tight.
Last I would swap the starter relay with another like relay.
Do each step one at a time so as to find the correct solution.

 

I had this exact same issue. It would not engage the starter, or it would partially engage and begin to crank and then fall out. After repeated attempts it would catch and start. Once the engine warme'd up it never happened. It was only on cold mornings on the first cold start.

I was leaning towards the solenoid, but why only when it was cold? It was not battery related, nor was it the battery cabling. Swapping starter relays had no impact. I never resolved it, and being in Texas it was an infrequent occurrence.