Your assumption is correct, and Google is wrong (sortof and maybe).
The possessive pronoun ihren is definitively accusative singular.
However (this is a bit off-topic, because it's more about English than about German): While German does not really care that many boys maybe use many kites (and can thus use the singular, even if it's most probably many kites), English is a bit more precise here and requires you use plural when each boy or girl has their own kite.
You could say there is no way to translate this sentence literally, assumed the kids do not share one single kite.
Astonishingly, Google is clever enough to get the pronoun right when you ask it to translate a sentence that makes it explicitly clear there is only one common kite:
Die Kinder ließen ihren einen Drachen steigen
the children let their kite fly
Die Schüler sollen ihr Englischbuch rausholen und die Seite 23 lesen.
is a perfectly acceptable German sentence, and every pupil will assume to read their own book.
Pupils are asked to open their textbook and read page 23
Would be considered wrong (or rather, irritating) in English because pupils are assumed to each have a book of their own.