It a little difficult to tell, but this is most likely Common chickweed, Stellaria media, although it may also be a white flowered variant of Scarlet pimpernel, Angallis arevensis. If it's the Scarlet pimpernel, the stems should be square when you roll them between your fingers and you'll find spots on the undersides of the leaves.

Common chickweed is a winter annual that germinates summer through fall. You can easily manage these by lightly tilling the soil and disrupting their root systems. You can read more about both of these plants here:

https://plants.ces.ncsu.edu/plants/stellaria-media/

https://extension.umass.edu/weed-herbarium/weeds/anagallis-arvensis

Consider using a mulch material in your garden to prevent further weed growth.

https://www.canr.msu.edu/resources/smart_gardeners_improve_their_soil_and_weed_control_with_organic_mulch

Thank you for your input.  I do not think this is chickweed.  The pimpernel comes closer; the leaves look similar to what I'm dealing with, but the flowers on my mystery weed are white and much, much smaller.  Thus I'm not sure we are there yet with the ID of this weed.  Any further ideas you might have would always be welcome.

Ok, that's helpful about the flower. I agree, it's been a tricky one for me to put my finger on since many of the characteristics are just not a good fit for the most common weeds we see in a veggie garden. Are you able to give me some more site information? and possibly send some pictures of the entire seedling (including roots) placed on a contrasting background along with a ruler or coin for scale? 

How long has it been growing at your site? Did you have any plants nearby that could have gone to seed in this area? What natural features are near your garden? lowland, upland habitat? How much sun does the area receive? Anything else that might be pertinent.

Thanks again for your reply.  My vegetable garden is 10 by 20 feet and is situated alongside the house, with a south and west exposure.  The adjacent house has a neglected, weedy area just to the south of my garden, but there has never been any sign of the weed in question growing in there.  I have been working this same garden for over 40 years, and for at least half that time I have routinely spaded grass clippings and maple leaves into the garden in the fall in an attempt to lighten and improve the soil.  For better or worse, I have tried to keep oak leaves out of the garden - even as we have many large oak trees in the yard - in the belief they would make the soil too acidic.  Despite these efforts, the soil remains quite heavy.

I don't really have a great way to receive text messages. If the photos are on your smart phone, and you are able to pull this message up on your web browser, you can add the photos right from your phone library using the "choose them" option in the little blue box. 

Alternatively, you can email the photos to yourself from your phone. Download them on your computer and attach them that way.

In the meantime, I'll keep thinking about what this tough little plant could be. Thank you for the additional info about your garden site, that's very helpful!

My colleague gave me a recommendation for this one. We may have found our answer with False Pimpernel Lindernia dubia. We were close!

https://michiganflora.net/record/1638

https://www.minnesotawildflowers.info/flower/false-pimpernel

You might notice that it is adapted to bare mud or wet sand on shores, river banks, marsh and pond margins. I wonder if your veggie garden is close to a body of water as well. You said the soil is heavy, and I wonder if the water table is quite high there.

If the soil stays moist for prolonged periods this could impact seed germination and cause dampening off of seedlings. We see this most often in a greenhouse setting, but I'm wondering if what you noticed this year could be related to soil moisture. https://extension.psu.edu/damping-off

Thank you for your continued work on this!  I am in the process of trying to send you more photos as you requested but have hit some technical obstacles at this end.  I will keep trying.  After looking up lindernia dubia on Google, the one small photo provided comes the closest yet to the weed in question, although there is no scale in the picture, and the flower looks a little big; the flowers I'm seeing here are very tiny.

The plants can be as small as 2 inches with flowers that are 1/3-1/2 inch in size. Smaller plants may have smaller flowers depending on the growing conditions.

It's possible the weed seed was brought into your garden at some point, but it's hard to say where it may have come from.

Best wishes with the garden going forward!

Thanks for the info.