Answer: The veronica may be spreading by both creeping and by seed and in that case it could blow in more easily than you could spread it with a mower. Unless the chopped bits are large enough and land on bare ground, odds are that it will not be able to root after mowing. Usually weeds infiltrate a lawn area that is not as healthy as it should be; the sparse lawn fails to compete with the weeds. This can be caused by poor soil, poor choice of grass variety, poor mowing/maintenance practices, or insufficient light.
Digging out the lawn and replacing it with herbs in an interesting idea, however do keep in mind that many herbs need full sun and good drainage to do their best. They will also require an initial establishment period and many are also attractive to deer. (Unfortunately, in my experience the only solution to a deer problem is a fence.)
Some herbs to consider for morning sun and a rich moist soil would be mint, sweet woodruff, or lily of the valley all of which would spread as groundcovers. For a drier soil you might consider basil, parsley, and chives. If your soil is well drained and not too heavy, you might also try the traditional culinary herbs such as sage, oregano, thyme and so on and see how they do -- they may prefer more sun than they will receive there but it is always worth an experiment.
Good luck with your project!
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