I have a 2006 diesel X Type which was a nightmare for starting when the engine was warm. No matter how cold the morning was the car would start on the first turn. Drive it 2 miles to the petrol station to fill up and the thing would take ages to start. Quite embarrassing.
I had it Terra-cleaned for £120 and the problem was solved straight away.
No need to do anything mechanical.
She is on 172000 now and drives nicely.