"One Hero may use a Power now"?

they get an additional power. Any time that phrasing is used -- may do ____ now -- it does not count towards the limit on how many you can do per turn. That's why you can chain together all 3 of Tachyon's Fleet of Foots, Hud Goggles, Pushing The Limits, Lightspeed Reflexes (play 2 cards now), Lightspeed Reflexes, and still be able to play 3 more cards before your play phase is up.

The only exceptions that come to mind are

A.) Vengeance Blade's Impulsion Beam, which SPECIFICALLY limits each hero to 1 play and 1 power per turn, no matter what,

and

B.) The standing rule that you can only use EACH power once per turn. So unless you have a Gauge or Seg Fault or something in play already, you might want to give the power to someone else so you CAN do something during your power phase.

When you play a card that lets someone use a power and target yourself, you're still within your Play phase as you haven't yet finished resolving the card you just played - assuming there are no other instructions after the "use a power now" part of the card, said card will be resolved once your power's effect has finished. So were you to use such a card on a fully set up Adept, you might be waiting a while for it to finish ;). Anyway, once that's done, then it's your Power phase and you proceed with that as normal, and like McBehrer says, you still only get to use a specific instance of a power once per turn so if you used your played card on yourself to use a power in your Play phase, you won't be able to use that same power now it's your Power phase. Obviously, if you have two copies of a non-Limited power-granting card out and used one off the one-shot, you're free to use the second copy's power in your Power phase, though :).

You don't ordinarily get to use "one power" in your power phase you get to use "a power". If something was limiting you to only 1 power each turn then Hasty Augmentation or Between the Lines would use up that 1 power limit for the turn, but there is no such limit in the game ordinarily and HA and BtL do not apply such a limit.

If played on their respective characters they could still use a power during their power phase, just not the same power that HA/BtL had let them use (that is to say, not the same power from the same card- the standard limit the game applied to all).

You get to use one Power right then and there, and you may use another one during your power phase as normal, but you cannot use the same Power both times (from your character card, or from one particular Ongoing or Equipment card in play - a second copy of the same card is a separate Power even though it does the same thing).  Each Power may be used no more than once per turn, and the normal number of powers you can use in a turn is 1; Hasty Augmentation grants another 1 to that total that turn.  If you get a power usage when it isn't your turn, however, you can use the same Power you used on your last turn.

Note also the difference between card effects which tell you to skip your power phase (does not prevent using Hasty Augmentation to at least get one power that turn) and ones which tell you that you can't use Powers (does prevent it).