I actually think this is important.
He actually moved in with her for about a week and then came home.
So basically they had an affair. Both to blame on that one. Affairs are never right but it takes two etc. However he then promised her he would leave you and he did. This made her trust him.
He then MOVED IN to her home, presumably with her kids. Allowing this is a massive step for a family. He then says "oh sorry I changed my mind" and runs back to you a week later.
How do you think her kids felt about this? Of course the kid knows about the affair since your husband MOVED INTO THEIR HOME. However brief his stay presumably the kid was told he would be living there, they would be a family etc.
Did it ever occur to you that he has been an absolute asshole not just to her but to her kids as well as to you? He's basically stomped all over two women and two sets of kids because he can't keep it in his pants ffs.
The gift could be:
Her kid misses yours. They had play dates etc and could have built up a friendship. Don't forget her kid also thought your families were going to be joined. If this was suggested over a long period of
Time they could have begun to see your kid as a new sibling. You don't know the relationship between them through the older kids eyes.
She wants to send a message to your partner that she was accepting your kids (and him) into her family and she's still here.
She wants to piss you off so you kick him out.
Personally I think that whatever the reason for the gift he's much more of a threat to your happiness than her since he feels it acceptable to treat people this way.