Having done a search, there is a lot of information on various sites inc CAB saying that if you have a child together, the person who owns the house cannot 'evict' the other person and that they (the non-owner) can apply to the courts to allow them to stay for up to 6 months.
It's great you've done a search.
You appear to believe that either the mother can be forced to live with her partner against her will for up to 6 months, or that an 8 week old baby has to move out with the mum for six months, or that the mum has to leave her baby with the father at this age, just so he can keep on living in the house.
Do you think that is reasonable? Or do you think the law is there to protect the child's right to a home? Because I can assure you that that is the intention, and therefore, it is inapplicable here. The child's home us not under threat.
And if you are referring to an Occupation Order, those are most commonly used in situations of domestic abuse, in order to allow the victim to remain in the home. As far as we know, OP is not abusing her partner, and they are specifically stated in the legislation as to be used only as a last resort, because interfering with property rights is a serious matter.
I can't offer advice here. I'm not qualified. I have worked for the CAB, and have an LLB and LLM, but this is a complex area of law. I only know enough to know that you do not know enough. If you could advise competently after a brief swirl on a search engine, nobody would need lawyers.
what is the point discussing how much he pays? 
Because under the law, if you contribute to the mortgage monetarily you have a claim, potentially, on some of the equity in the house. It can be argued that there was an intention for it to belong to you both. All depends on alleged intention when you started to pay, which can be tricky to establish after a split, with nothing in writing. And similarly, if you pay rent, then you have more rights than if you just defray your own daily expenses.
If it were a grand a month plus, and that helped towards the mortgage, he could argue he was intended to share equitable ownership (though if it was a whirlwind, and the baby a newborn, he's not been there long enough to make much of a claim, I believe?). And if it was agreed as rent, then potentially (although I really don't know) he'd have more claim now, too.
Surely this was an amount agreed when they were happy together, as a reasonable contribution. It's not our place to query their arrangement now. Maybe the OP is a millionaire and she only wanted some money from him towards food.
Sure. But there are legal implications, potentially. Which makes it relevant.