What's more important in a difficult room, room correction or higher, clean, power?

I have had room correction based subwoofers and they are limited as to what they can correct in the bass response.  
They definitely do not satisfactorily correct issues that are caused by poor fundamental integration between room and main speakers.  
Fundamentals include best position in room for main speakers, best position for subwoofer, proper subwoofer crossover frequency and phase and proper subwoofer level.  
Once these were set well the sound was excellent and I could not tell any difference when room correction was engaged.  
My recommendation would be to get some RTA software (phone, laptop or tablet) and a CD with full bandwidth pink noise and start measuring the response and then adjust positions, crossover, phase etc until you have a smooth response.  It is always comforting to know your response is smooth.    Then "season" to taste, slight adjustments in level to suit your preferences. 
  

Room correction was the most important factor in my tweaks, and easily the most dramatic improvement of my audio system ( I am in a small room and use reflective and absorbing surface in some ratio+ cheap schumann resonators+ many Helmoltz bottles homemade) ...But before that it is very important to heavily controls the vibration resonance problem linked to all the gear and speakers....After that room treatment will be possible and astounding if you make it happen with your ears open...In my case the cost was low because I only use homemade or cheap products, and even with that the change was epic toward holographic sound and imaging and natural timbre of instruments...

1+ Ralph. As Ralph says no amount of digital correction will fix a standing wave. For most of us you have to use multiple subwoofers. This does not mean that room control does not have a place. The best room control will provide perfectly flat frequency response across the entire audio spectrum and assure that the response from both channels is exactly the same. This gives the best image. From there you can modify the frequency response to taste and do some very cool stuff. My unit has dynamic loudness control. It automatically jumps from one Fletcher Munson curve to the next with changes in volume. It is programmed with 8 Fletcher Munson curves which can be modified to taste. The end result is that the spectral balance remains the same from -40 dB to 0 dB. Bas management is insane. You have independent control over high pass and low pass filters at 1 Hz increments from 0 to 375 Hz with slopes from 1st to 10th order and you can change it all on the fly. The unit will keep nine frequency response curves in memory which you can switch by remote control. My normal curve slopes down from 1 kHz to 6 dB down at 20 kHz. I have another curve down 12 dB at 20 kHz for bright records and another with a 3 kHz notch filter for sibilant records. There is no way you can do these things in the analog world and for those of you worried about resolution this is all done 192 kHz 48 bit.    

What's important (as others have said more verbosely) is to fix the acoustics. It cannot be done with more power.

IMO it is best done by First, positioning listener, main speakers, and subs for smoothest response; Second, adding bass trapping and 1st-reflection treatment if necessary; and Third, using DSP if you want to smooth really low frequencies, say below 80 Hz, further.

In doing all of that, measurement capability (e.g., REW and a calibrated mic) helps a lot.