Here's why:
* Prepositional phrases start with a preposition (in this case, "with") and are followed by a noun or pronoun (in this case, "a sigh").
* Clauses contain a subject and a verb. "With a sigh" does not have a subject or a verb.
Example:
"She looked at him with a sigh."
In this sentence, "with a sigh" acts as an adverbial phrase modifying the verb "looked."