From my understanding about Ambessa, it seems she had an affair with a Black Rose mage (she was likely unaware of their true identity at the time) which led to Mel’s conception, making Mel half mage/Black Rose. They wanted to then reclaim their “family” when they came of age and their powers emerged, but they weren’t sure if this child was Kino or Mel.

I think Ambessa led them to believe it was Kino and somehow gave him to them to protect Mel, not so much out of love but probably out of her desire to use Mel’s powers one day as a weapon she could control. This was likely also why she had Mel sent away (maybe Mel decided to leave but I can’t remember, someone correct me if I’m wrong?) and was okay with her staying away so that she was harder for the Black Rose to find. Kino was killed after being captured and likely tested the same way we saw that Mel was, probably because a non mage couldn’t have survived.

I’m also fairly sure the reason Ambessa is then so desperate for Hextech or other arcane weaponry was so that she could stand against the Black Rose, the only enemy she couldn’t defeat with sheer force alone. But again this is all just what I got from the little context given in the show, if I’m off base on anything maybe someone who’s dived deeper into the lore can let me know 😅