1. She kept her promise to visit her aunt regularly.

2. She kept her promise that he would visit her aunt regularly.

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Are both the same in meaning?
And is 'her' promise' in apposition with the 'that clause'?
In Sentence 1, does 'to visit...' modify 'her promise'?

There are no appositives here. #1 is better wording than #2, but both are grammatical. "kept" is the verb. "promise" is an adverb telling what was kept. "to visit" is a prepositional (adverb) phrase modifying "promise", telling what the promise was.

See your previous post.

1. She kept her promise to visit her aunt regularly.

She - subject
kept - main verb
her - possessive pronoun, modifying "promise"
promise - direct object
[to visit her aunt regularly] - infinitive phrase that could be considered either an adjective phrase, modifying "promise," or an essential appositive. The adjective use is more likely.

2. She kept her promise that he would visit her aunt regularly.
She - subject
kept - main verb
her - possessive pronoun, modifying "promise"
promise - direct object
[that she would visit her aunt regularly] - noun clause. Since noun clauses cannot be used as adjectives, then this must be an essential appositive to the word "promise."

Noun clauses, appositives, and essential/non-essential elements in English are not always easy to understand or to identify!!

Both sentences convey a similar meaning, but there is a subtle difference in emphasis.

1. She kept her promise to visit her aunt regularly.

In this sentence, the focus is on the action of visiting her aunt regularly, which is the promise she made. The phrase "to visit her aunt regularly" functions as an infinitive phrase that explains the purpose or goal of the promise.

2. She kept her promise that he would visit her aunt regularly.

Here, the emphasis shifts to the fact that she kept her promise. The phrase "that he would visit her aunt regularly" functions as a subordinate clause that describes the content or details of the promise.

Regarding the second part of your question, in both sentences, "her promise" is the subject and main focus of the sentence. The phrase "to visit her aunt regularly" in sentence 1 and "that he would visit her aunt regularly" in sentence 2 provide additional information about the nature of the promise. They do not function as an appositive because they are not restating or renaming "her promise."

In sentence 1, "to visit her aunt regularly" can be seen as infinitive adverbial clause that modifies the verb "kept." It explains the purpose or intention of the promise.