The correct sentence is “He has already gone to school.” It’s in present perfect tense.
Let's consider the grammar points governing the sentence.
First, both “has” and “have” are auxiliary verbs; they've got no difference in their meanings and both of them are used to make present perfect tense. The difference lies only in their usage.
It’s a grammar rule that “has” always follows a subject that’s third-person singular, including he, she, it or any singular or uncountable noun. On the other hand, “have” is used only with a plural noun or pronoun, with the exception of “I” and “you.”
Here, the subject is “He,” which is third-person singular. So, it has to be followed by “has” and not “have.”
Second, in forming present perfect tense, both “has” and “have” are always followed by the past participle form of a verb. “Went” is the past form of the verb “go.” So, the question of using “went” after “He has already” doesn't arise at all. It has to be “gone.”
Besides, it must be remembered that one of the most important differences between simple past tense and present perfect tense is that we don’t specify the time of an action with present perfect tense; whereas with past simple, it's always indicated. When it’s not specified using any time expression, it’s assumed that the listener knows the time of the action or can make it out from the context.
Here, in the given sentence, the time of the action "go" is not indicated. Moreover, the emphasis is on the result of the action rather than on the action itself. Therefore, it would be grammatically incorrect to say "He already went to school." Instead, we should say, "He has already gone to school."