"What does the child eat?"
Translation:Mit eszik a gyerek?
Eszik is the 3rd-person singular indefinite form, yes. The form that you find when looking up the word in a dictionary. (It's also the 3rd-person plural definite form, but that doesn't apply here.) Eszik is one of the so-called "-ik" verbs ("ikes ige") which all get an -ik as the 3rd-person singular indefinite suffix.
"Esz" doesn't exist in Hungarian.