Is there essential difference between "Focus on what you want
to have happen" and "Focus on what you want
to happen" ? There's a difference, but not a very strong difference.
If yes, is it an expression of strong desire? In a way, yes, I suppose, because the idea of controlling your actions in an effort to reach your goal is involved.
What you want to happen is your wish. It may happen; it may not happen. You accept fate.
What you want to have happen is your goal. You may succeed in causing it to happen; you may fail. You attempt to control what happens.
On the other hand, what you want to have happen can be taken to be, as you say, your very strong wish or hope. You wish you could influence the world and the people in it so that you get what you want. In this interpretation, you put yourself in a more dependent role. You're depending on others to arrange things so that you get what you want. The control is indirect or perhaps even missing.
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The versions with the past participle are simply the passive variant of the same structure:
I'm going to have the plumber fix the pipes. (Active)
I'm going to have the pipes fixed by a plumber. (Passive -- with agent)
I'm going to have the pipes fixed. (Agentless passive)
CJ