"I have children."
There is a rule of phonetical softening in Polish ( I am not sure if it is proper grammatical term.) If there is an 'i' after c, dz, z, s, it is pronounced as ć, dź, ź, ś. Threfore dzieci is pronounced as if written (dźeći). The letters dzi and dź are pronounced the same. The other letters(ci, ć...) are pronounced also the same. Dzi and dź is similar sound as j in jeep, ci and ć sounds simmilar like ch in cheek. Important thing for pronountiation is create these "soft sounds" on your front teeth.
Could someone please explain the pronunciation of "dzieci"? Isn't "c" stands for "ts"?
Why would "Ja mam dzieci" not be an acceptable answer, when "Ja jem" is grammatically acceptable? Does it have to do with the possession of the child versus the action of eating?
"Ja mam dzieci" emphasises that "I DO have children", so unless you want to emphasise it, omit the pronoun. Nevertheless, "Ja mam dzieci" is grammatically correct, report it next time.