Learn English – “on leave from” vs. “on leave on”

The leave them kids alone part is definitely ungrammatical, as is the first chorus line We Don't Need No Education (the former should be those kids, and the latter uses the much-reviled double negative).

Per this answer on a related ELU question, using "them" as an "article" is non-standard (but not uncommon in speech, particularly with the less well-educated, or in "facetious" usages).

There's nothing ungrammatical about leave us kids alone - if there were, you wouldn't expect to see written references to things that might help us writers, for example.

In such constructions, the [optional] "class/category name" after "us" applies to all of us - it doesn't select kids or writers from a larger group. The context inherently implies that "we" are all kids, writers, whatever.

I "have to" take a leave (or, probably more common leave of absence) is present tense and can indicate "compulsion against the speaker's will."

I "had to" take a leave is past tense and can indicate the same thing.

I "have had to" take a leave is present perfect tense. It follows the normal pattern of forming the present perfect by using have + particle of the verb. In this case the participle of have to is had to. So you get have + had to.

In your sentence, the present perfect means that the leave you were forced to take started in the past and is still in effect at the moment of speaking the sentence.