What do I do?

My partner and I separated after I decided to keep the baby. Been together for 6 years. Have three kids already. Never planned for a 4th but neither did we declare that abortion would be a way. Aside of the pregnancy he suddenly realized the weight of the partner who is not working (we don’t have any family or friends who could help and I’m still in school for and AA) and wanted to go.

He found himself with another woman in a few weeks after declaring he didn’t want to be with me anymore. Moved out a month later. I had to sign an agreement where he takes kids for three days (or he wouldn’t pay the rent).

Now he is taking kids for once a week and they just stayed over last night. All this time I never spoke to the person he’s with.

My oldest child is heartbroken by the situation but willingly spends the time with her dad and his woman and her daughter because she has a lot of fun with the girl. Just to be clear for the time we’ve been together he rarely spent time with kids or helped at home. His idea was that he was paying for things and that was enough.

Now after the fact of about a month of them hanging out together he says I need to speak to the person he’s with. In my culture this is no no and an extreme disrespect to a person. Anyone been there? Is it reasonable? What should I do? What do I tell her?