Note: Wait i made a mistake, i am using a Zed-10FX so the picture is wrong. I need to change it.
Ahh, ok cool, i did get confused when you said there were internal effects as i didn't remember seeing them but on the updated photo they're clear as day
In which case, yes you would use the "Aux Out" to the TC-helion. Tbh i don't know the desk but
on some on a stereo channel if you plug something in to one of the inputs it will come out in mono. Alternatively you could just plug the output of the TC-helion into one of the "line in's" on channel 1 or 2. So, you could plug the two mics in to channel 3 and 4 (M3 and M4 on the desk), the "Aux Out" of the desk to the input of the tc-helion, and the output of the TC-helion into channel 1 or 2.
On the desk, each channel has a "strip" of different knobs from top to bottom, in this case a three band EQ, then the built in FX knob, then the aux send, then pan, and finally output volume. The way i find to imagine what an "Aux Send" does is to compare it to a tap in your house: imagine the signal from the mic travelling from top to bottom down the channel strip. This signal is like the mains water for a house. The "Aux" knob on each channel acts like a tap to take a part of the signal away from the mains and send it off somewhere else. If we imagine that the mic plugged in to channel 1 is "hot" water and mic on channel 2 is "cold" water and you want to run a warm bath you could turn the hot tap all the way up and the cold tap on just a bit. The result is a lot of the hot water goes in to the bath (or, in our audio case, the bath is our "Aux Output") and a bit of the cold tap does as well, but the mains water carries on going where it was before (or in this case the main signal from each channel). To complete the analogy, if we want the warm bath water to mix back in with our mains we pull the plug in the bath and it ends up in the same place as the original mains water (or, in terms of audio, our "Aux Output" goes into whatever device we are sending it to, and then plug that device into a separate channel on the mix so that it can reach the main outputs alongside the original signals)
I hope that makes sense
So, in essence, by turning up the aux knob on a channel it will add more reverb/delay to that sound. You can do this with numerous channels and only one reverb/delay by sending different amounts of each channel to the aux send.
For karaoke i'd imagine a reasonable reverb would be more than enough seeing as the channels already have a three band EQ