In phase 10, can I pick up a card from the discard pile and then relay that card back down in the discard pile?
Yes. The rules say that you must draw a card and that you must discard a card as part of your turn. (At least my edition) It does not state specifically about not being able to discard the draw off the discard pile.
However, I do question it as a strategy. If the card to be picked up from the discard does not help the hand, then there is a much greater probability that the card from the draw pile will help the hand.