Speedomter cable noise

Your car does not have a speedometer cable it has an electronic speed sensor that sends an electrical signal to the speedometer head. Could the noise you hear be a cracked exhaust manifold, the noise is more of a fluttering sound on acceleration perhaps worse when the engine is cold. I have seen a lot of cracked exhaust manifolds on the ES300.

You will need to get a mechanic to remove the speedometer head and possibly replace the whole odometer Head, hopefully you can only replace a cable or a part where it attaches to the Head.