"They would like children together" makes sense--in its most common form, it describes a commonplace prospective family situation. I can think of no logical reason why "mutual children" would not carry the same meaning--but that's not what we'd say in English, and it does sound very weird.
I'm not sure that the German sentence makes any more sense than "mutual children" does - or that "they want children together" is a correct translation - since gemeinsam is being used as an adjective for Kinder (the -e ending makes this clear). See Jolutti's (top-level) comment below, also.
"Sie möchten gemeinsam/zusammen Kinder haben." - here stays the rare possibility open that the children are maybe not the biological children of both of them.
here 'gemeinsam' is better than 'zusammen'. wir möchten zusammen schwimmengehen. 'zusammen' is in my language feeling more like two people have the same wish and do it together. 'gemeinsam' is two people have the same wish and wish to do it together. I don't know how I could explain it better.
PS: be careful Duolingo said: Sie möchten gemeinsame Kinder.