I saw this come up as a bug for the Video Game (although, getting stuck is a bug), so I wanted to mention the rule here. If a card grants "an additional power this turn", it allows you to use an extra power during your power phase. If that card is played outside of your turn, you don't get a power use, because you have no power phase. So, if a card like Smite the Transgressor or Flame Spike is played outside of your turn, you don't get to use a power.
This came up during development and we could not find any official ruling. It is working as we discussed with Christopher. Here is the logic, which also generalizes to "play an additional card" and "draw an additional card" text:
If the card granting an additional power use is played on a turn other than the card’s owner’s turn, then they immediately get to use a power. If they do not do it immediately, it's gone.
If the card granting an additional power use is played on the card's owner's turn, after the use power phase, then they immediately to use a power. If they do not do it immediately, it's gone.
If the card granting an additional power use is played on the card's owner's turn, before the use power phase, then must wait until their use power phase to use the additional power. (this is the typical use case, BTW)
Great, your version is better than mine! 
I suggest you both read Spiff's excellent FAQ which includes the ruling on cards which grant powers outside of your regular power phase (And nothing goes into Spiff's FAQ without coming from an official source), it accords with MigrantP's version (as it would since they both come from Christopher).
It’s on my to do list.
Spiff's FAQ is something we refer to all the time, and it does not cover specifically what Pydro is talking about. We also searched the forums/bgg and could not find anything about it.
People forget how literal a game Sentinels is.
Also that 0+1=1.
But yeah, I've always played it as MigrantP describes, makes for some fun first turns when Omnitron-X hits Sentinels with his power and they top deck Team Communication or Unique Capabilities.
It really does, I wouldn't have mentioned it otherwise*. The only thing not covered by it is that you lose your 'extra' power use if you don't use it NOW (outside your turn).
*I even double checked before I posted.
Can you give me the page number where it clarifies that "you may use an additional power this turn" and "you may use a power now" work the same way outside of your turn? Or how it works during different parts of your own turn? Because that's what this is about.
Pg 24. Bottom, under Overdrive. If played out of turn you still do what the card says, and may use strike twice that turn. That was the ruling that clarified out of turn power use.
As a courtesy it is probably best to reference where in Spif's FAQ a ruling is, since 48 pages of rulings is pretty hard to just remember or even search through.
Esp. because Spiff has properly kept things like the Overdrive ruling, that has global implication, attached to Overdrive where most people might not think to look.
That does not exactly cover the phrase "additional power this turn" but I can see how it could be interpreted to apply. It does not cover the other situations I mentioned above, too.
It says "this turn", so it doesn't have to be your turn ;).
From what we've been told the power phase is just the time in your turn you get to use a power by default, that is all.
The ruling from Spiff also shows the current standing on when those powers occur. Since you play Overdrive in your play phase, and can use a different power, then use strike twice, the powers granted by card play don't have to be immediate to the card being played. I remember that being discussed, and I believe there was an official ruling on it, but searching really old posts is a nightmare, so I can't be 100% sure myself.
As for being granted an additional power in a turn, The rules of Arithmetic apply, 0+1=1. If you get 1 additional power that turn, and you normally would get 0, you now get 1. Unload works like Overdrive, Expat gains a number of power uses that turn equal to the number of gun cards in play. As Christopher has stated before, you aren't limited to using powers, drawing cards or playing cards to the specific phases of your turn, those are just the times that you get to do that action naturally.
These are older rulings, which is why they don't have their citations, but they are from official word. Things are always subject to change, and I can definitely see extra powers and cards being granted occurring immediately, since otherwise it would lead to a complicated interface for the player.
The way you describe it above seems like the best method for doing things in the video game, and while it is a slight alteration from current official word (on these forums) things change like that a decent bit, and gameplay wouldn't be hurt by it.
Related to the topic and about the video game implementation, I was using Ra against Omnitron in the Ruins of Atlantis and the Atlantean Font of Power came out, playing Flame Spike from the top of Ra's deck...
and then the game froze up--It told me to select a power, but I wasn't able to select a power. Ra's thought cloud appeared (displaying his cards), but when I tapped on the cards, nothing happened. I used the rewind button several times, but the same thing kept happening.
Same thing happened to me (I reported it on the Handelabra site). I was able to get out of it by double-tapping on the Flame Spike card in Ra's thought bubble and then using the left/right arrows on the zoomed card to go to the power I wanted to use, then I could click the "Use Power" button.
MigrantP said this may be fixed in 1.1.1
Were this the case then I should have been able to use the power on The Staff of Ra, either.