Yes it will cause the noise. The reason it goes away is the metal both manifold and head heating up and expanding to fill the small gap between the metals.
A fix is the removal of the manifold and hopefully the bolt left the shaft extending beyond the manifold. If the bolt is exposed it makes removal much easier, If not then it has to be drilled out and either tapped or extracted. Not an easy job on the vehicle but sometimes can be done. If there is no room to work on the bolt to be extracted by drilling then the head has to be removed.
In either case unless you do it yourself, it won't be real cheap. Several hundred dollars to a grand is my guess at 120$ per hour depending on the solution to remove and replace the bolt.
An other point I forgot is the type of motor. Since you did not mention if you have a V10 or Diesel it will make a difference on fixing it. The V10 has aluminum heads and the diesel cast iron. Both has their pros and cons.
