"– Are you crying? – (sob) I am homesick."
Translation:– Itketkö sinä? – (nyyh) Minulla on koti-ikävä.
Perhaps, but no one would practically ever say it without sinä without any previous context.
You would only hear it in something like "I think I will cry now. - Well, do you?" "Luulen, että itken nyt. - No, itketkö?"
Reported: "Itketkö? (nyyh) minulla on koti-ikävä" should be accepted.
What I understood is that the 1st and 2nd person sing. and plur. can be left out. In Finland they always leave them out. Only in cases that you would like to emphasize that specific I, you or we, it is added. In the contrary, the 3rd person sing. or plur. needs to be in the sentence.