Final answer:
The infinitive phrase 'to fetch a pail of water' in the given sentence serves as an adverb modifying the verb 'went' by explaining the purpose of the action.
Step-by-step explanation:
The infinitive phrase in the sentence 'Jack and Jill went up the hill to fetch a pail of water.' is 'to fetch a pail of water.' This phrase is being used as an adverb because it explains the purpose or reason why Jack and Jill went up the hill, thus modifying the main verb 'went.'
It gives more information about the action by answering the question 'Why did Jack and Jill go up the hill?' to get a pail of water. The construction of the infinitive phrase includes the infinitive 'to fetch', which is the base form of the verb preceded by 'to', followed by the direct object 'a pail of water.'