The gap follows the preposition "with" and the quantifier "no". In English, prepositions are followed by a by a noun phrase. A noun is a word that can occur after a "noun-marker". Here, "no" acts as a determiner/marker. “Intention” is a noun meaning purpose or aim. The phrase “with no intention to…” (or more correctly, with no intention of…) is a standard English expression used to show that someone did not mean to do something. So the sentence means that her laughter was not meant to ridicule the applicant.