Ok, lets think big picture here, the whole truck. When the clutch is engaged, the torque twists the driveline in relation to the body, toward the passenger side. Meaning drivers side of the engine moves upward toward the hood. So as the truck is traveling down the road the engine, transmission and anything attached is being twisted toward the passenger side. When the clutch is pushed, the driveline stands back up, moving everything toward the drivers side.
The left hand turn lifts the drivers side of the truck. A left hand turn makes the body lean toward the passenger side, effectively raising the drivers side.
So, to me the common point is the drivers side being "raised" in respect to the driveline. Clutch in, or hard left makes the gaps on the drivers side larger. Or moves something on the passenger side move away from what is causing the noise.
What exactly is causing this, I don't know. But it could be something that is rubbing when the torque pushes it up into something. Or something screwy like the exhaust pulling on a mount in a strange direction and removing the engine torque or twisting the body and frame in a turn can make the pulling stop. I've chased noises over the years and there are times I've had to take this approach, as what is really making the noise is on the other side or end from what I think it is......
Also, just remembered this, check the throttle cable routing. I remember someone on here chasing a noise in the cab, and it was the throttle cable resting on an injector line and telegraphing injector and engine noise into the cab.
