You can't have a second copy on board. Once you have claimed one, you can't launch another, make progress at your opponent's copy or claim it: it is scored only once.
Your mothership has +1 initiative.301.3, 302.2a, 701.2a, 304.5a
While this card is on board, your initiative is 1 higher, and several such cards add up. The higher initiative gets the first chance in each window, dispatches last and assigns all battle damage. It changes right away: if this card leaves the board in the middle of a sortie, the rest of the sortie follows the new initiative. Once the expedition is claimed, it works from the logbook for the rest of the game, for the player who claimed it, whoever launched it: "you" is then the claimer (304.5a).
Distance 10 · Spoils 3304, 706
Crew dispatched here on their own turn make progress, whoever launched it; on the opponent's turn they can only fight (706.3a). The first player to reach 10 progress claims it and gains 3 spoils.