Only if you have a thing that lets you.
The order of (play, power, draw) is fixed but there are effects that let you play a card or use a power 'out of sequence'.
For example, Omnitron-X (in the Shattered Timelines expansion) has a card that lets you draw a card at the start of your turn, and a card that lets you play a card and use a power at the end of your turn.
If you had both these cards out, your turn could look like this:
* Start of hero turn: Electro-Deployment Unit lets you Draw or Play
* Use a power
* End of hero turn: Slip Thru Time lets you Play and Use a power
So your turn could start with Draw (which is usually the last thing you do), then Play, then Use a Power, then Draw (again), then Play (again), then Use a Power (again, but note that each instance of a power can be used at most once a turn).
OmX can get much crazier than this once he has a few cards out...