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

Re the OP’s scanario:

"Tryin’ to get high, without having to pay..." -Marianne Faithfull

Room correction is not a fix, the effect of the room problems still exist and would still be interfering. It’s like a continual hasty pasting over of an ongoing audio murder.

Room correction can be done judiciously, minimally...and then one has to figure out if the given minimal corrections are worth the problem of interference in signal purity that is brought about via the installation and use of the hardware the creates the corrections.

Then to understand such under the given signal disturbances and coloration that is carried out by the choices in the rest of the gear that is in the room.

physical acoustic correction via physically correcting the room is the best bet, every time and likely always will be. It is just the nature of how acoustics and audio works... that makes active correction suitable for non fidelity situations like concerts, malls, airports, bars, commercial theaters, etc...places where fidelity is not the primary concern.

If DSP was the fix that it is advertised to be by some, then it would have taken over the high fidelity world by storm.

Even though it has been around for quite some time, note that it has not taken over.

Thus, thinking cap..a bit of funds...some work..and fix acoustics.

IF this cannot be done, then the room must be accepted as it is, or try some minimal DSP correction, with the caveat that the carrier of the DSP, the digital systems involved - are destructive to overall fidelity.

One can fix some of the annoying bits with the DSP and then find themselves listening less and less to music in that space, as the positives are outweighed by the negatives of the digital manipulation. Negatives that take time to consciously discern and put label to.

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...

The room modes below 50Hz were best avoided by careful speaker and listening position placement in my room. Corner traps helped somewhat along with panels.

Dirac Live with the filter on still sounds markedly better. Mixed phase filters with phase time alignment and magnitude correction creates vivid, lifelike soundstages. The usb signal exits the PC digitally corrected, so no additional A/D anywhere. Agree that deep nulls caused by room modes are not solved by room correction nor are first refllection points, but what about everywhere else? I'm so happy I can flick the Dirac filter switch to on and let my ears decide.

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.