What are the popularity news cards for?

They are exactly what the name implies; they're used to increase the popularity of any particular news headline story that you wish to make more popular. Every story has the little flame icon where it has a numerical value. Higher the number, the more popular it is. Usually you use them when any story appears that involves you you your faction members. Like when one of ya'll reaches mansion level 25 or level 60 underboss or unlocks T4 associates