@Hachiman you sound incredibly stressed and I wonder if a chat to your GP might help, you might want the chance to talk to someone about the financial pressure you are under.
I agree with all the other posters your son is better off with you, his mum and his sister.
I say that as a parent with a birth daughter and an adopted son.
IMHO you need to write down the suggestions here, evaluate them honestly and go through step by step.
Asking advice 're benefits, work, etc, talking to your wife about how stressed you feel. I would imagine giving up your new born baby will not be on her horizon so I think you need to be very careful what you say to her.
My friend is currently in very dire financial problems. They are heavily in debt. She has managed to talk to a debt management organisation, a charity, which has advised her. All the family are getting jobs to help, they have looked at their budget for food and stuff, it has been very tough.But they are altogether and would not be apart.
Honestly, the best place for your son is with his family. But you need help, advice etc, and support.
Will your wife be able to work in the future? Do you have support from your own parents/in laws/siblings etc?
You are not a bad person, you are a brilliant dad who really cares. It does come across in your brief post. You are articulate and caring.
And it goes without saying - look into the contraception thing quickly.