Desk layer keeps moving with body

Since the desk is nested, the "fixed" pins will fix it to the parent mesh (not the scene) so will not achieve what you want unfortunately. Instead you can put a stick across the top of the desk if you want, but if the desk is independent it should not warp anyway (at least, after removing the pins).

The problem is the independent layer is still attached to its parent mesh (the rest of the character). So if the attach point moves, the desk will move. To reduce this, you can pin the character to the scene, then attach the desk to the character where that pin is (or below the pin). Some sticks across the character above the attach point can also help the body move less near where the desk is attached.

The other thing to try is change the "attach mode" for the desk layer. There is hinge, weld, and "free". I am hoping "free" will solve your problem, but have not tried it. I did the other approach, it worked well enough for me, so stopped looking.

If you search through the forums for "desk", there have been a few other threads on exactly the same problem. They might have some useful hints or suggestions too. But I think having the desk in the middle of the puppet artwork is the best approach if you want the character moving behind the desk a bit. The problem is just making sure the attach point of the desk to the puppet does not move, and pins on the desk won't work in this case.