Answer:
influence
Step-by-step explanation:
In the sentence "I have an in with a member of city council who keeps me informed." The word "in" indicates that the speaker has an influential relationship with some member of the council. This influence allows the speaker to have access to some information to make sure what is going on within the board and what consequences that information can bring. For this reason, we can conclude that the correct answer to your question is "influence".