When an Imperial player spends influence to purchase an agenda card, and that card is discarded after use or after a round(s), does the player get the Influence they spent on the card back?

For example,an Imperial player spends 2 influence to buy an Agenda card that costs 2 Influence. He now no longer has any influence points left. That particular card instructs that after the round ends, the card is discarded. Does that player now have 2 influence again after he discards the agenda card? Or is the influence spent gone for good?

It's gone for good.

Discarded cards go back into the box for the rest of the campaign and those influence points are spent and gone.

Spend them wisely.