Dazed condition, and having an effect from a token take the last health before power phase

[Hordemonger]

So if I have a character that is holding a cube that will make the character suffer one damage in the cleanup phase, and they have one health before  the cleanup and then suffer the one point of damage to flip their card does that mean that for the duration of the next turn for my team that character is out? I’m basically asking if I have a character that is sitting on a cube that makes them take damage and they have a would left before the cleanup phase, they then take the damage which would make them flip their card at the beginning of the power phase so they would be in able to activate for the remainder of that Teams then? 

[Negoldar]

Cosmic Cube Fragments damage the character holding them during the Power Phase, not the Cleanup Phase.

In your described scenario, this will result in the character being Dazed in the Power Phase and remaining Dazed through the entire Activation Phase and into the Cleanup Phase. During step 4 of the Cleanup Phase they will flip to Injured.