Do the members of a married couple with children have a moral obligation, not (just) to each other, but to their children, to not cheat on each other?

Parents certainly bear many moral obligations to their children, including obligations related to how the parents interact with each other. That said, I don't think that monogamy is morally required of married parents or other parents who live together in a sexual relationship. To the extent that an extramarital sexual relationship could be carried out entirely separate from the family -- and from family responsibilities -- it might be simply irrelevant to parenting. Perhaps many individuals do not have the skills or are not in a situation where to make that separation possible, but some parents may have those skills or be in that situation. Likewise, I suspect there are parents in loving relationships, and who are effective parents, have chosen to reject monogamy and who have also learned how to construct a good family life for themselves and for their children. For more on this, the discussion by the self-described "kinky" communities. So, monogamy may be a useful practice for many...