In this sentence the word irritable means, according to the Cambridge dictionary, "to become annoyed or angry very easily". The fact that this child has not slept the amount of time he needs to sleep makes him angry.
However, you should always resort to other sources so as to be sure. Similarly to the Cambridge dictionary, Collins indicates that this adjective means "to be easily annoyed". Yet it offers the following synonyms to replace the word: bad-tempered, cross, snappy and hot.
