@Satinacarpark how are you today?
I have been where you are (with your P, my children were both younger so not the same as your 9 year old) and I'm sorry to say that in the end - not immediately but eventually - the relationship ended. With my P that is, not with my children.
I didn't succeed in getting him to respect me and it was not a tolerable house for me to live in under those conditions. I was really struck by the PP who said "It's your house! go home and read the riot act!" YESSS. I understand totally why you were crying in a car park on a winters' night with your baby, rather at home getting some rest in your own space with your home comforts about you - I totally get it, there was no room for you there - but it is WRONG that you work so hard for your family and are not respected enough at home to be able to cope.
But - reading the riot act. You should be able to say to your P "this is not ok and this is why and this is what has to change." but what has to change is that he has to respect you, and I don't know if you can make a man respect you who does not already. If he doesn't see why you deserve his respect, love and support, I don't know what can make him see it. I am not saying it is not possible; I am saying I couldn't do it with mine so I don't have the advice to give you on that.
I live alone with my daughters now and I am very happy that I am not modelling my partner treating me like shit in my own house, and me putting up with it. I work very hard, and my kids perceive this and respect me, and I do not have to fight for respect or decent treatment in my own house. My kids are 11 and 9 and while they can be moody and difficult, they basically respect me. (for now!)
the biggest issue I have with their behaviour is related to too much time online. I am zero tolerance on quasi addictive behaviour with screens and will turn the whole lot off when I have to.
Your P is the issue. You deserve better.
Good luck.