IF (key word) the problem is related to the loss of valve lash due to thermal expansion of the metal in the engine this can create an engine miss and possible engine damage, all depending on which valve (or valves) is tight, how long it’s been going on, etc.
IF (again) there are no engine codes present and the fuel pressure is fine then I would run a compression test on a warm engine. If there’s an abnormality on a cylinder then that cylinder should be retested with the wet method; meaning a small shot of engine oil is added to the cylinder and the compression is retested.
If the compression does not rise much that can point to a cylinder head valve seating problem, which means a lash and/or valve burning issue.
Sorry I can’t be more precise but there’s a lot of blanks as far as the info goes.
I’m having a very hard time seeing this being a mount problem because if that were the case it should not be temperature or time dependent.
One thing that is quick and easy to do before running a compression test could be to connect a vacuum gauge to the intake manifold. A vacuum gauge will reveal in seconds if there’s a tight lash issue, timing belt off a tooth or two, clogged catalytic converter, etc.
Hope some of that helps anyway.