There is a lot of question of what aux does for any given power, so here I will list the effect auxiliary power has on any given power. Most of this information came from going into the game, and seeing what happens to the power information when I adjusted my aux power settings.
This information should be accurate as of season 3.
This power gets significant bonus from aux power, and you probably want to use this with as much aux power as possible.
This power gets a very situational or marginal bonus from aux power and/or it may not affect the effectiveness of the power itself or its major effects.
This power gets no bonus from aux power, or the bonus is too small to be meaningful.
Auxiliary power does this for the given power:
Charged Particle Burst: Increases shield drain
Energy Siphon: Increases buff/debuff duration
Feedback Pulse: Increases damage reflected
Gravity Well: Increases damage and repel strength
Hazard Emitters: Increases heal and damage resistance
Jam Sensors: Decreases fragility
Mask Energy Signature: Increases stealth*
Photonic Officer: No discernible effect
Photonic Shockwave: No discernible effect
Polarize Hull: Increases damage resistance
Science Team: No discernible effect
Scramble Sensors: Increases duration
Tachyon Beam: Increases shield drain
Tractor Beam: Increases damage
Tractor Beam Repulsors: Increases repel
Transfer Shield Strength: Increases shield healing and increases shield resistance
Tyken's Rift: Increases damage and power drain
Viral Matrix: No discernible effect
Auxiliary to Emergency Battery: Increases power buff
Auxiliary to Inertial Dampners: Increases resistances and speed/turn rate strengths
Auxiliary to Structural Integrity Field: Increases Heal and Resistance
Science Captain Powers
Photonic Fleet: No discernible effect
Science Fleet: No discernible effect
Sensor Scan: Increases resistance debuff
Starship Dampening Field: Increases resistance buff
Subnucleonic Beam: Increases recharge time debuff
Increases stealth detection rating on science ships
Increases stealth detection rating on non-science ships
Increases cloak effectiveness*
Decreases carrier hangar recharge times
*Cloaking mechanics are notoriously hard to test, however the magnitude of change when changing aux power settings was insignificant compared to the magnitude of the stealth granted, according to the tooltip.