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If you put cotton mittens on the baby, it'll stop scratching its own face.
Literal
Cotton [possessive-の] mitten [object-を] put-on-[if-ば] baby [topic-は] own [possessive-の] face [object-を] come-to-not-scratch.
Practical-advice register. The conditional ば takes care of 'if you do X.' Notice 自分の顔 — 自分 is the reflexive 'own/self' that refers back to the subject of the clause it appears in (here, the baby). ひっかかなくなる stacks ひっかく ('to scratch') with the negative + なる pattern: 'come to no longer scratch.' The compound verb ひっかく itself is built from 引く ('pull') + かく ('scratch') — a common 'pull-and-X' compounding pattern.