Quick answer: No. Cashing out an entry that has a promotion applied forfeits that promotion permanently. This is true no matter which promo it was, how much value it had, or why you chose to cash out—there are no exceptions.
Why the promotion isn't returned
Cash Out settles your entry immediately at its current fair value and closes it out — the entry is done, and nothing about it changes after that point. A promotion (like a No Sweat Token, Gimme Pick, Boost, or Discount) is tied to that specific entry running its full course. Once you cash out, the entry no longer plays out normally, so the promo tied to it doesn't carry over, isn't refunded, and can't be re-applied to a new entry.