i found choosed minor power card by aid from lesser spirits is immediately put into play(not play), and can target by repeat effect.
then, at the end of turn, that card is discard to their personal discard pile? or, that card is still stay in play?
and, when spirit gain a major power card, can forget that card?
You have three questions there:
(a) I believe you're correct, that the card from Aid from Lesser Spirits can be targeted by Repeat effects. I'm not 100% positive.
(b) The card remains in play next turn and all future turns.
(c) No, the card is not one of the Spirit's powers; you can't forget it.
dpt is correct, according to the living FAQ. In addition, the elements on the card are immediately available and will stay available for as long as the card is in play, e.g. the rest of the game.
Yup, it just goes wherever you put your "in-play" cards and then stays there forever, with no way of discarding it or anything like that.
I belatedly added a FAQ on the second point.