Can I eliminate some background noise on live recording?
It depends on the kind of background noise, and how loud the background noise is. See: https://manual.audacityteam.org/man/noise_reduction.html
…on live recording?
Audacity will only do it later. Audacity doesn’t do anything in real time. If you need real time processing, some recorders and most smartphones have the ability to live record with voice processing, echo suppression and noise reduction.
Of course, then your show is going to sound like a cellphone. Environment processing is what does that.
Good live field recording is not for the easily frightened.
Here’s one a bit more modern. That’s from one of the Matt and Tom videos. They’re literally on the park bench but the voices are terrific and perfectly clear because they’re using a radio-connected shotgun microphone with a dead cat windscreen. That’s probably $2000 USD right there—that Matt’s holding with one finger!