Importunate means annoyingly persistent or relentless, like a cranky child's importunate demands for candy in line at the grocery store checkout.
Importunate rhymes with unfortunate, and importunate questions, in their irritating persistence, are unfortunate indeed. Importunate restaurant patrons tend to inundate their unfortunate waiter with demands, and one student's importunate questions can keep a teacher busy through an entire class period. You'll see in importunate the Latin root port, which means "harbor." From this came importunus, "troublesome," but also "having no harbor."