The camera is a Panasonic HC-X920; a consumer model HD camera from 2013. It's a good little camera and age should not be a factor. It has a mini (1/8 inch) audio input, and they sell outboard "shotgun" mics that one can attach with a mic shoe attachment. So in theory, external mics are supported.
https://www.panasonic.com/mea/en/support/product-archive/camcorder/HC-X920.html

I bought a Movo LV1 lavaliere with a mini jack to use with this, because all my other XLR lavs are XLR. But connecting this mic creates a hum that cannot be rectified by changing the battery or making sure that the battery is right-side-up. Neither does using a TRRS to TRS adaptor
https://www.movophoto.com/products/lv1-usb

Using a Sony 44B lav with an XLR - mini adaptor (which worked just fine on my very old Sony Digi-8 camera) enhances the audio with a very similar sounding hum, although of lower volume. A different XLR - Mini adaptor did not fix it.

Sending the mic signal through a mixer and testing mic and line level outputs did not resolve the problem.

But a cheap ($10) plug-in power condenser mic (with the TRRS to TRS adaptor) solves all issues.
https://www.amazon.com/dp/B07PTPMGFH/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20/din02c-20

So, problem solved, but I want to know why. I was sick that day during microphone school, and need the right words that explain why this mic works and real mics don't.

Also, and more importantly: Is there an adaptor that will correct the issue so I can use any microphone or connect it to a mixer?

Any help appreciated. Thanks.

(Did NOT expect an Amazon logo advertisement to replace the link to the product page. Pretty clever, Jeff.)

It seems odd that a TRRS to TRS adapter would be needed, but perhaps it just happens to put the right mic plug contacts to the right socket contacts. I'd put a meter on it and find out exactly what goes to what. Have you tried the cheap mic without the adapter?

Most likely there's some configuration of adapter that will allow an XLR mic to connect to the input jack, but it's most likely not simply going from the three XLR pins to three TRS contacts. Perhaps this:

https://www.bhphotovideo.com/c/product/1321775-REG/kopul_cmx_1_v2_2_channel_mixer_with_shoe.html/BI/2855/KBID/3801

But I don't think that provides phantom power.

I suspect that the camera mic input is set up for a stereo mic, tip-left, ring-right, sleeve-ground. I would have expected the cheap mic to interface directly with that without an adapter, with its signal going to tip and ring (for dual channel mono).

It's a $13 mic off Amazon going into an old consumer camcorder.

Not sure why but a couple thoughts...
That lav mic you tried has 4 contacts and the camera jack is only going to accomodate 3.
The camera mic jack is providing power to the mic, which might be incompatible with a mic that uses its own battery, such as that lav mic does. Hate to ask this but, did you try it with the lav mic switch on and off? I think the off position cuts off the microphone power (same as Rode Video Mic Pro).

The Panasonic compatible mic is the VW-VSM10.

This annoys people who buy mics for Sennheiser radios too - 3 circuits, let alone 4, so so many different standards for what goes where. Audio can be tip or ring, DC can be tip or ring. DC can be on the same circuit as the audio, or on the other. Move to 4 circuit connectors and multiply the randomness.

In my case, I want a 3.5mm locking plug. I see adverts, and there's about a 33% chance it's going to work.

The simplest way to fix it is to get a plug, stuff it in the socket and check which pin produces the hum - that's the audio circuit. Then meter the tip and ring and see which has 5v on it, using the sleeve as return.

Then solder it to either the mic cable - or better still, solder it to a 3 circuit socket wired differently that lets your mic work.

Sorry for the delayed reply. I had to dig the mic out of the trash to test something.

Re: TRRS: Many mics that are aiming for the cell-phone market are TRRS, but my camera mic input is mic only, so I need a TRRS to TRS adaptor. (My newest cheap mic purchase came with an adaptor) But I did check the Movo mic without the adaptor anyway; no good, as expected.

I had also checked the mic with the power switch in the "Off/SmartPhone" position, and it was just dead. But I had not checked the mic without any battery in it! I did. It didn't matter; did not solve the problem.

I do not have a meter, and will not take the time to tear apart the wires to effect a solution.

My biggest mystery, though, is not really the cheap mic that doesn't work. Why wouldn't my Sony 44B with an XLR-mini adaptor (that has been used in prior applications) work? As I mentioned, this has worked fine with other (older) camcorders. When I use it, I do hear sound from the mic, but there's a hum, the volume of which changes when I hold the cable or let it hang.

Also: two different XLR-mini adaptors, same result. So its not one bad cable.

Thanks for the insights so far.

The adapters or plug configuration are likely shorting out the bias voltage (aka, Plug-in Power.. not to be confused with Phantom Power which is a different animal).

The Panasonic mic that's sold for that recorder is the VW-VMS10. A photo of the mic shows the word "stereo" on the label, and a TRS plug. Since there are just three terminals and two audio channels, that mic has unbalanced audio. The wiring standard for such a mic was Tip = Left audio (with DC bias voltage fed from the camera), Ring = Right audio (with DC bias voltage), and Sleeve = common ground and cable shield.

This wiring standard was used on a lot of consumer and prosumer equipment. For example the Tascam DR-07, DR-05, DR-03 recorders use it, Philips DVT7500 (and most of their pocket "voice" recorders) use it, Alesis recorders used it, etc. etc. etc.

The same standard is used on many consumer stereo mics, such as the AT Pro-24, AT822, Sony ECM-737, and many others. Presumably, any of those mics should work with your HC-X920.

The Movo LV1 has a TRRS plug. Right away you should be on your guard. The TRRS standard, devised by the Apple Empire, is evil. It is a very real source of confusion and problems. A TRRS mic has the hot mic signal on the sleeve (which should be ground/shield in normal equipment). The TRRS mic has the shield on R2 (the ring closest to the sleeve) and that connector doesn't even exist on a normal TRS connector (like the one in your camera).

A mic wired for a TRRS phone won't work with a TRS recorder. BUT Movo has some sort of inline switch box, which supposedly makes it compatible. However, I can't find a description of the wiring inside. So we have to make some assumptions. One position of the switch is labeled "OFF" and the other position is labeled "Camera." I assume that means in the Camera position something is ON (opposite of OFF). Presumably in the Camera position, the battery is in the circuit. What does it do? What is the battery polarity? Are you sure the battery is inserted correctly? Might the battery in the switch box interact with the DC bias coming from the camera? There's no easy way to answer these questions, since Movo doesn't seem to provide enough technical details.

Regardless of the switch, there will still be a problem with the cable. The mic has a TRRS plug. With TRRS, the shield is always on Ring #2. But your camera wants shield on Sleeve. So when you plug that mic into your camera, you essentially have a piece of unshielded cable in the circuit. That's the source of your hum. Also the DC bias polarity is reversed. With TRS the sleeve is negative. With TRRS the sleeve has the positive bias voltage.

My suggested test is this: put the Movo switch in "OFF/Phone" position, so the signal on the Movo plug will hopefully match the TRRS standard. Then try the TRRS/TRS adapter that came with your other mic, and insert that between the Movo plug and your camera socket. That might work, depending on how the adapter is wired. Without documentation about the innards of the adapter, it's always somewhat a crapshoot.

A better solution, IMHO, is this: If your recorder/camera wants a TRS mic connection, buy a mic with a TRS plug. Leave the "smartphone" mics and accessories for use with phones.

As far as your Sony 44B, I'm keeping this in a separate reply, to try to minimize confusion. Again, the problem most likely arises in the adapter. There are three terminals on the TRS, and three terminals in the XLR. Therefore there are theoretically at least six possible ways the adapter could be wired inside (actually a few more). So an adapter configuration that works with some of your older cameras won't necessarily work with the HC-X920. Without seeing schematics of the adapter, and of all the cameras involved, I can't be more specific than that.

Again, you don't have a meter and don't want to take the time to analyze this yourself. So just accept the fact that there are many combinations and some of them won't work. (You can't put Yugo wheels on a Dodge truck ... does it matter why that's true?) IMHO a proper adapter would include an audio transformer to convert the mic's low impedance (~200 ohms) to the recorder's impedance (probably ~2,000 ohms) and also convert from balanced to unbalanced signal. That could be a bit pricey, and, again, you might get into the jungle of connector adapters.

If you're not happy with your $10 mic (that works correctly) there are better mics made for TRS connections. My first look would be the Rode GO lavalier, or the Audio Technica line, but there are certainly others.