This is the noise right at 5.6 seconds in the right channel, correct? Yes, every 1000 Hz. Here's what the SPAN spectrum looks like (right channel in red).
Yes, you can filter it out by creating notch filters for each peak. I use Reaper ReaEQ plugin (free).
To me, this sounds like cell phone interference. I'm not sure if the newer phones cause problems, but I've heard plenty of this on my older recordings when I was using the 2-wire, unbalanced cables that come connected to Church Audio mics.
At a minimum, try to recreate the issue with your phone and if that's a source, make sure your phone is off when recording.
Turning the gain up or down on the recorder probably won't have much of an impact since the noise is getting into your mics and/or cables, before it hits the preamps.