I am sorry but where in Rings of Power does Halbrand ever mention having a child?In this series, Sauron supposedly had a child, whom Adar killed, which is definitely not from Tolkien's legendarium. I don't really care what Amazon makes up for the Second Age as long as they don't pretend it's canon. What's been shown so far doesn't fit with what Tolkien published in either the Appendices or The Silmarillion. To me, it's just expensive fan fiction. I wish it were a lot better, but I fear we're going to get more of the same. I'd like to stop paying Amazon money, but I need them for other things, so I'll just enjoy the thought of Bezos uselessly burning money.
The only mention I can find even close is not made by Halbrand. When Hallbrand asks if Adar recognizes him, the answer is no. Hilbrand's expression is a little confusing here. But he doesn't appear to be pissed about the lack of recognition. But he then pull back to kill him. And only stops when Galadriel orders him to stop. And here is where he is struggling. Fighting against his own desires.
Now Adar sees that struggle and tried to push his buttons. Did I harm someone you cared for? A woman? Perhaps a child. We see Halbrand is still struggling, but he pulls back.
Halbrand has three interest here.
1. if Adar recognizes him his game is over. Secret out. Halbrand for that reason would be wise to kill him before anyone can question him. Or before any aspect of his behavior sparks recollection.
2. Adar (who to the best of our knowledge is fairly honest in his dealings) states he killed Sauron. And while the show is nor required to follow the text (as this is an adaption), Sauron can have his physical form destroyed. So Adar might have betrayed him and killed him. Meaning you might very much want to end Adar's existence.
3. And finally he is trying a different path, one where he wants Galadriel to be part of it, and to start his path to power in a different less direct way. If he wants to continue that path, and stay on Galadriel's good graces he needs to spare Adar's life at least for the moment.
Thats the struggle. To kill the being who destroyed (and betrayed) him. Are to pare him at least temporarily to stay true to his current path.
The child or the woman is just Adar trying to attempt a reason that would push a South Lander, who seems to know Adar into a killing blow.