You definitely want to enhance with your second prophet; if you use that one to spread your enhancer will cost 500 faith. So you're spending 800 faith to get 4 spreads and an enhance; you can get that for 700 if you go initial prophet -> missionary -> enhance -> missionary, without incrementing the GP counter. Plus missionaries move quicker; plus this way you get either the initial missionary or the enhancer quicker.
Really the question is, do you go initial prophet -> missionary -> enhance or initial prophet -> enhance -> missionary. I think most people favor the initial approach.
The only time I would consider using it to spread rather than enhance is if my core cities have been flipped to another religion.
But there aren't really any bad choices for enhancer
I would say it depends on more than just this. If my scouting revealed some religion crazy civ (like Ethiopia or Egypt) i'll expend first. Sure i might be third to enhance, but then there are enough good enhancers for at least 3 religions. I might also miss the second follower belief i wanted, but i can adapt to a sub-optimal 2nd belief. The best beliefs in the world won't help me if i can't spread my religion.Unless your generating a lot of faith or are intending on getting the enhancing Prophet for free via HS or Liberty Finisher, then it's skip the missionary and go directly to enhance to ensure you get that second follower belief you were planning on getting.
Enhance. If you want to spread early, use a missionary. I usually go Prophet(to found) -> Missionary -> Prophet(to enhance).
Once you've captured a holy city, you can use an inquisitor to purge its holy city status (that is an inquisitor trait that a Great Prophet does not have).
If other neighboring cities still have that religion, it may reappear in the captured capital through normal religious pressure, but its holy city status will not reappear.