"Juego con mi hijo."
Translation:I play with my son.
That would be “at my son." If you meant the personal 'a', it doesn't apply here. 'son' is not the direct object. A direct object is the noun which is being 'verbed'. 'son' is not being 'played'.
A good indication that the personal 'a' is unnecessary is the use of the word with/con. “(verb) con a..." doesn't make sense.
Refresher: In Spanish, the verb conjugation often gives you such an excellent indication of person and tense that the subject is known without being explicitly stated.
Juego must be first person, singular, indicative present tense. It can't be used in any other way. In English, first person singular is 'I', never 'you'. The 'Yo/I' is left off in Spanish because the verb tells you that it has to be 'I' you are referring to. Translation into English usually necessitates an explicit subject, because our verbs don't give all that information.