If your system hums, teach it the words: “In a recent project with a church choir, the best take had an air conditioner hum in the background. I had the group sing with the track already there in hopes of beefing up their sound and covering the hum. They sounded great, but the hum was still there. Is there a way for me to sum the sound (a hum) on two different tracks and phase cancel it out?”
I assume you used stereo miking techniques. A mono signal would be easier to deal with. The only thing I can think of that is easy enough and has a decent chance of working is to make a recording of the hum, and only the hum, with the mics placed exactly where they were and with all settings exactly the same. If (and this is a very big if) you can make an exact recording of the hum as you recorded it with the choir, you may have decent results by mixing a polarity reversed version of it back in with the choir mix.
There are also hardware and software devices designed for removing hum and noise from signals. Roland has made several very effective hardware units over the years. Digidesign‘s DINR noise reduction plug-in for Pro Tools works well and Sonic Solutions has a system that works well. While you may not want to invest in any of this hardware or software yourself you can probably find a recording studio nearby who has something that can do this (our studio can do it if it comes to that). The results are never perfect, but often are good enough to make a recording useable.