Can a player discard the objective card already played near the player board to activate this function?
The rule doesn't particularly point out that it should be the objective card from the hand.
Well, there is this text on page 10
Thee most frequent requirement is discarding cards. Wherever an action shows one or more cards with a red arrow, you must discard the exact depicted card(s) from your hand to your face-up personal discard pile in order to gain the reward shown next to it. (As mentioned before, you can perform the action only once, no matter how can you can meet the requirement.)
Additionally, page 15 says:
Note: You are not compelled to play your objective cards. If you want, you can keep some or all of them in your deck by the usual means (for example, by discarding them via this icon (draw one, discard one icon)). For each objective card that is still in your deck at the end of the game, you may then decide if you want to score it or not. For the cards placed in your personal objective area, you do not have that choice. Those must be scored.Page 16 describes the final scoring process as
decide whether she wants to remove it from the game or add it to her personal objective area now (without using its immediate action).
I hope that helps. By the way, is there a reference to discarding objective cards that were placed in your objective area? I haven't found any, but perhaps I missed it.