It seems that I am missing a relay from the fuse and relay box that sits behind the battery in the trunk. Plus the relay that is in place for the fuel pump does not look original and seems to not really fit the socket very well.

Anyway the missing relay seems to be some accessory relay. Can anyone tell me what sorts of things it is supposed to work for? I.E. what things might not be working if this relay is missing?

See the attached photo.

Many....not all, but many...of the relays are rather generic an can be replaced with off-the-shelf items. And that appears to be the case with your fuel pump relay.

Poor fit? Can you be more specific?

If the relay pins are not snug in the socket terminals this can typically be corrected by using 'something pointy' to compress the terminals a bit so the relay pins engage more firmly



It is quite common to have vacant wiring connectors and relay sockets....almost certainly for some equipment that your car doesn't have. The Jaguar diagram simply calls it an "accessory relay" (as you have done). It is probably used when xxx-type of equipment is installed to satisfy the requirements of a certain market and/or for equipment installed on xxx-market cars, but not others, for sales/marketing reasons.

Someone will likely come along with a more specific answer

Cheers
DD

The relay was just barely making contact in the socket. I pulled it out with barely any effort and then re-inserted it with a firmer push into the socket. Once I did that the car fired right up. I have been driving it all around today and it seems fine except for the "clear" transmission fluid that I obviously have to drain & refill with proper stuff.

It really felt like the plastic case of the aftermarket Hella relay was sort of fouling on the edge of the socket and therefor perhaps not really fitting completely down into the Jaguar relay socket.

Doug is indeed correct the missing relay is the "accessory relay" & is shown on the attached diagram. The cabin accessory connector is behind the side heelboard on the left hand side. (bottom of the A pillar)
The relay is actuated with aux. & ignition on thus providing power to this outlet.
On my vehicle I have fitted a suitable relay & use it to provide power through to my caravan when attached. Power to the relay comes from fuse F13 in the trunk relay box.
You are not missing out on any item working if no relay fitted, it all depends on what you connect to the trunk or cabin accessory connector.

John Herbert