cut-elimination provable in PRA

My suggestion is Theorem 5.17 in section V.5.(d) of Hájek and Pudlák's Metamathematics of First-Order Arithmetic, Springer-Verlag, 1993. There, cut-elimination is shown to be provable in the even weaker theory $I\Delta_0 + \text{superexp}$.

I think the chapter "Proof Theory: Some Applications of Cut-Elimination", by Helmut Schwichtenberg, from "Handbook of Mathematical Logic", might be what you are looking for.