He went back with the purpose to see his family one more time.
He went back to see his family one more time.
He went back for the purpose of seeing his family one more time.
Use to + infinitive to express purpose in normal everyday English: ‘He went back to see his family.’
The more formal expression for the purpose of is followed by a noun or the -ing form: ‘The meeting was organised for the purpose of discussing the new plans.’