Thanks HWYSTR455.
I found a picture which shows the back of a clock, and I think I found the answer to my question. There is a spring with a gear pressed onto the back of the clock adjustment stem. It looks like the spring keeps the gear away from the clock gears. In order to adjust the clock, it's necessary to pull the adjustment stem outward (against the spring tension), which will allow the adjustment gear to engage with the clock gear, so the time can be set. The gear probably fell off my adjustment stem. I'm hoping the gear and spring are still inside the clock and I can just reinstall them.
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