We. The Revolution

I mainly want to warn that influence points appear to reset from day to day, so any influence points you don't use after ending the day will be wasted. There are also certain events or factors that either decrease the number of points you receive or prevent you from earning them altogether (also not sure if this is a mechanic or a bug yet).
You get influence by:

• 3 points automatically each day
• controlling districts: 1-3 points depending on district controlled (denoted by the number on the district)
• filling out over half of the report correctly: 1 point
• successful negotiations: ? (the rewards sometimes show the influence icon, but I'm unsure if it actually works)

Places I've found where you can spend influence points are:

• asking for mentor's assistance with questions: 2 points per revive, can only be used when too many mistakes have been made
• asking mentor to help fill report: 1 point per question, cannot be used past a certain point in the game
• reveal emotion during speech/negotiation: 1 point per hidden emotion
• sending diplomats to negotiate: variable, only done in certain intrigue actions
• plaza building actions: 2 points for spending time with family, 4 points for facilitating negotiations, 5 points for top building actions, 6 points for all other building actions
• random events: variable depending on options during event