how to stay in China for long term for me and my son. My partner will work there.

You can only obtain a tourist visa with 30 days stay duration because you are not married, you and your son apply for tourist visa as if your partner doesn't exist, you need to get married to obtain a long-term visa, once you get married, you and your son can apply for a Z visa as accompanying family dependents of personnel working in China.

You should provide proof of kinship, e.g. marriage certificate, birth certificate and a copy of the Z-visa as well as Residence Permit of the person working in China.