"in possession of a competence" in this case is meant to mean that she came into a modest amount of money, either lump sum or ongoing, that would either support her in modest circumstances or make her a more attractive marriage partner to a man who doesn't need his wife to have a fortune but would still like her to make a financial contribution to the marriage.

Some people who need heirs, but aren't necessarily upper class, find a healthy, living child a better indication of being able to produce of the same than never having had a child.

You found some new typos for me. Thank you.