Problems with one car, not the other

Fuses inside the caravan? ...... 🤔

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work via xc90 so I would suspect socket on LR. you may have voltage at socket but not connecting clean to plug on caravan. spray plenty WD40 on socket and plug. push in and out several times so contacts rub against each other and try again see if works.

Try putting a bulb across the socket a voltage might show, but have no power

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Thanks all. Great help. Will try these things.

If it is a bad earth return, is that the earth where the towing kit electrics are earthed to the car? I did take that connection off, clean it and refasten ensuring it was a good snug fit under the bolt.

Someone offline suggested the towing kit module might have failed. Would I still see a voltage in the car socket in this case?

Looks like you'v got voltage but low currant, depending on how your electrics were wired in, if it was after market one you will have a fused power source from the battery to a relay in the boot, you would need to check the connections for corostion or a bad connection this would includ checking the earth, if it's a Land Rover factory kit again it could be bad connection,