The scenario I'm imagining: a Han man and a Han woman living in an urban center get married and have a child (let's say a healthy son so there's no chance they'll be able to get special dispensation to have a second child). Man and woman get divorced. Man marries a second woman who has no children. Can they have a child? Or does the child from the first marriage 'count' as the product of the second marriage also? What if the sexes are reversed, and it's the woman with a child who gets remarried to a childless man? If either of these scenarios were allowed to have a child, was one more likely to end in state-sanctioned pregnancy than the other? Would the answer change if the first spouse had died rather than been divorced?
And to that point, was sterilization ever common after the birth of the first child? Like, would the man get a vasectomy to ensure that he couldn't have a second child? Or would the woman have a tubal ligation? Or would they just use contraceptives, abortion, and optimism to keep from having a second child?
Or is this assuming it was a far more defined practice than the policy actually was?
I'm on mobile so I any give a link but if you dig through my post history on /r/asksocialscience you'll see a lengthy post about the one child policy. When I have a chance I'll dig for you.
However the short answer is that, as of the 1992 revision to the family planning policy, I the first child is legally registered as a child of the father post divorce, he is legally not allowed to have a second child. If a widow and a widower marry, each with their own kid, then that's obviously legal and there is no issue. But if a person has a child already, wen I divorced, the law says that cannot have another. They cannot adopt and they cannot have another biological child without the applicable penalties (fines etc)
I'll try to hop on an add the links and answer any other related questions when I have a chance but it'll probably take a day.
Also you can find the current laws on the issue online (if you can read Chinese) if you're interested. They're also pretty clearly worded, though sometimes somewhat oddly specidic( e.g. Single man can adopt female child iff they have a 40 year age difference).
And to that point, was sterilization ever common after the birth of the first child? Like, would the man get a vasectomy to ensure that he couldn't have a second child? Or would the woman have a tubal ligation? Or would they just use contraceptives, abortion, and optimism to keep from having a second child?
The least that I can do is try to tackle this part of your question (since I'm not sure as to how to answer the first part).
Urban couples were encouraged to have one child starting in 1979, when the slogan changed to "one couple should have only one child". Couples were encouraged to sign a pledge to only have one child. Following this pledge led to cash bonuses, preferential access to housing and daycare, and other urban benefits. Conversely, breaking this pledge led to reductions in salary and denial of food rations and healthcare for the extra child. The government ensured that birth control methods were made known to the general public, and women were encouraged to either get an IUD or to go on birth control pills. Birth control was considered to be in the realm of the woman, and she was in charge of procuring it and to use it. If birth control failed, abortion was always an option (but there was a stigma attached, since that meant that she failed to use birth control properly).
Source:
"Planned Parenthood"
Author(s): Honig, Emily.; Hershatter, Gail., 1988.
Source: Personal Voices: Chinese Women in the 1980's, pp. 186-191