It is so that the sound from the speakers lands directly into the microphone, but it would be too much effort to change the position of the speakers, and the microphone can also not move so easily for technical reasons...
So it's *acoustic* feedback then? Not much to do for that, except to actually move the mic(s) and/or speaker(s), which you say are both a problem, or to add baffles to physically block the sound from getting from one to the other. You can't stop it completely, but you can force it to take a much longer path and lose more energy along the way. If you lose enough, then maybe it's okay.
It may even be enough to put up just one flat, hard sheet to block the direct, straight-line path. Kinda like "shy baffles" for a drum kit on stage. Those are the clear plastic circles close to the cymbals, and their entire purpose is to keep the cymbals from drowning out the vocal mics.
Also consider the pickup pattern(s) of the mic(s) and the dispersion pattern(s) of the speaker(s). Put things in the nulls (directions of much less sensitivity), that you don't want to talk to each other. For example, a concert mic on a stand will usually start out with its null pointed directly at the wedge monitors if there are any. If you can change the pattern to something more favorable, by flipping a switch that some mics have or by replacing the mic even if it sounds good, that can help too, by putting the null(s) somewhere that you can use.
---
Or, you could turn off the speakers entirely, and use headphones instead. If you don't like that look on camera, maybe use some discreet earbuds as cheap in-ear monitors?
IEM's, as the industry calls them, and almost everyone LOVES them! They keep the stage quiet by replacing live speakers, which keeps things clearer for everybody, and they hardly ever feed back. They do sometimes bleed into the mic if they're not seated well in the ear and if they're turned up enough to probably hurt you if they were. So you're not *completely* immune, just because you're using 'buds.
(I've had that happen several times as a combined Front-of-House and Monitor Engineer: "Why do I hear the click track (metronome) in the audience?! I'm not sending it there! Oh! It's your ears getting into your mic.")
IEM's need to be managed a bit differently from live speakers too. They practically replace your hearing, rather than add to it like a live monitor does, so IEM's need pretty much a complete mix of *everything* that you want to hear, including ambient if that's important and so you might need to add mics for that, whereas a live monitor speaker ONLY needs to fill in what you don't already hear without it. Anything more than that is "mud" for everyone else.
