Since you guys know these cars better than myself, maybe you can help me figure out what this could be. My XJR makes a thump when accelerating from a stop. It's enough where I can feel it through the floorboard. It won't happen if I'm pretty gentle on the throttle. Maybe it's related, but I've also noticed the engine seems to move more than I would consider normal if I free-rev it. Engine or trans mount maybe? The right engine mount looks fine, I can't see left because the supercharger is in the way. Ideas?

 

Transmission mount is a common failure. All the old rubber on these cars is a source of trouble.

 

Check the four nuts securing the prop to the gearbox, have seen a few loose recently

Bit of a pain to get to on your back, but checking/ tightening nuts is free!!

 

You're pretty much going to have to find a way to inspect the
left engine mount. It is the more likely to fail given engine
rotation.

There are also the v-mounts and less likely, the rear subframe mounts.

 

There is a way to look at the drivers side mount but you need a flashlight to shine the light down there.
Basically look at the intake elbow (the one that leads into the blower after the intake intercooler and point the flashlight down towards the middle of the engine. Might need to look around a bit but you'll see most of the engine mount and you can examine to see if the rubber is torn. Ive just had to replace the mounts on my XJR and looking down there is how I determined the mounts were bad. I was able to see seperation on the outer part of the rubber.