If I move my Irelia from one battlefield to another (after holding), and then start a showdown with another unit there, and the enemy uses Thrill of the Hunt to materialize his unit to other battlefield. We both conquer, and then I move my Irelia back to the other bf, does another showdown start and can I score another point?

Yes, a showdown starts when you move Irelia back to BF A — but no, you cannot score another point there.

Here's the breakdown:

BF A is controlled by the enemy (they conquered it via Thrill of the Hunt), so when you move Irelia there, you're moving into a battlefield you don't control. That makes it Contested and a non-combat showdown begins (Rule 344, FAQs #10027, #11282).

However, Rule 465 says a player may only Score once per battlefield per turn. You already scored at BF A via Hold during your Beginning Phase, so winning this second showdown gives you control again — but no additional point and no conquer trigger (FAQs #7447, #7403, #5980). The same restriction applies to BF B if you already scored there this turn.