She tried to go get her hair cut, but her mother wouldn't let her.

Literal

She [topic-は] hair [object-を] cut [have-someone-do-てもらいに] go [tried to-ようとした] [but-が], mother [subject-が] [didn't-allow-for-me-許してくれなかった].

Two giving-receiving constructions in one sentence. ~てもらいに (purposive に on the stem of ~てもらう) — 'go (in order) to have (it) done' — chains the haircut intent to the going. Then 許してくれなかった uses ~てくれる: the mother is described from the speaker's (or 彼女's) perspective as failing to grant a favor. Without ~てくれる, it would be neutral '(her mother) didn't allow it'; with it, the slight injury is foregrounded.