wipe your mouth with 〇〇
(〇〇=例えば「ハンカチ」)
↓
〇〇をぼかすためsomethingに
↓
something to wipe your mouth with
(不定詞形容詞的用法)
↓
前置詞を前に出せばwhich or whomが必要(thatでは代用出来ません)というルールで
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something with which to wipe your mouth
となります。
a house in which to live
a friend with whom to play
という形と同じタイプのものです