"Whose child has a dog?"
Translation:Czyje dziecko ma psa?
"Czyje" is singular neuter or plural (but not the masculine personal plural) for "whose". "Kogo" is genitive or accusative for "who", and usually "whom" would be an even better equivalent:
Kogo zaprosiłeś? - Whom did you invite?
Do kogo należy ten pies? - To whom does this dog belong?
It is also a very widespread mistake among natives, to use 'kogo' when a form of 'czyj' should be used. If you really encountered 'kogo' as 'whose' in this course, frankly, I would be really surprised.