In this course, we only accept the following:
this can be det här or detta (and den här and denna)
that can be det där or det (and den där and den).
In real life translations, I'd say this: in some cases, the slightly more formal nuance in detta/denna may be more important than the this meaning. So I'm sure you can easily find contexts where detta can be translated as that. I also think these are all contexts where this and that can be used interchangeably in English with no difference in meaning.
I thought there was no difference between the "den har, det har, de har" and the "denna, detta, dessa," which you use is just which one do you like more. But now I'm seeing that there is a difference. That the Den, det, de ones are only for indefinite and the other ones are for indefinite.
Your phrase would translate directly to "this child hasn't mother". "Inte" is used only to form the negative of the verb in Swedish, so if you want to say "no mother" you are forced to say "ingen mamma"...
To be clearer:
Detta barn har en mamma=this child has a mother
Detta barn har inte en mamma= this child hasn't a mother
Detta barn har ingen mamma= this child has no mother