I work during the summer, therefore I am entitled to medical care (Tarjeta Sanitaria) but my partner doesn’t work during the summer, he labours in the winter on and off for a English builder and therefore is not contracted, or registered as self-employed, he has a NIE but not Residencia about two years he did have a contract but only for two months.

He doesn’t have a Tarjeta Sanitaria card. But he needs to have a cyst removed from near his eye. As it is not a emergency I believe his E111 will not cover this.

We are not married but we are having a child together, is there any way for him to get free medical treatment through the Spanish Hospitals?

If you are working and have a card all members of the household are covered providing they are resident and you are jointly padroned - he will also need to be registered with social security. So you have some admin to do but yes it is possible.

He doesn’t have residenca and we are not married but he does have NIE and social security. We rent our home, and are in the process of sorting out the empadronament, will that be enough?

Not being married is not a problem as neither are we, but I do have residencia and I think that will be a requirement - but rules can vary across different communities, so you will have to make enquiries.
They will probably require you to sign to say if the relationship ends then his cover is no longer valid.