Can’t figure it out
This looks like Houttuynia cordata, chameleon plant. This is a non-native invasive plant that spreads by its rhizomes (roots). We get many requests for help to get rid of this plant--which is still being sold by the nursery industry. It is extremely hard to get rid of and there are no easy answers.
If it is in small areas, you can hand dig or spot treat with the herbicide glyphosate as in Roundup. This is non selective and will kill all plant material, so you must be very careful to not apply it to the lawn or other desirable plants. It will probably take several applications to have any noticeable effect on the Houttuynia. You will then need to monitor the area for new growth and spray or dig as it appears. You must be sure to dig up all of these underground horizontal stems or they will sprout new growth from a small piece.
If the groundcover is in the whole lawn, you may just want to continue mowing and exhausting its reserves. It may end up being a good groundcover for dogs.
Total lawn renovation may not solve your problem, as you may have to battle it for years. Here are links to a couple of sites that discuss Houttuynia removal.
You can contact the National Pesticide Information Center for toxicity of glyphosate http://npic.orst.edu/