NOTE: I'm basing this on TFTD where X-COM's existence is publicly known from the start.
Each Nation has two parties: The Ruling Party and the Opposing Party. Through elections or violent take over the Ruling Party may change, along with the ones responsible for our paycheck.
Besides the invisible relationship meter already present in the vanilla game, called the Official Opinion from now on, the Parties have their own Personal Opinion of you. The parties will high opinions of you will try to raise as much funding as they can, and try to lower as little as possible when you screw up, and (in case Alien Infiltration are not instant-success) will not sign up with the aliens. Those will low opinions will try to lower the funding as much as possible, reluctantly raise as little as possible and sign up with the aliens without looking back.
The Parties also have two additional stats: Powerbase and Public Opinion; Both are essentially modifiers to the events and actions used by and against them.
So what if the one who doesn't like us is on top?:
1) You spend resources to improve their personal opinions;
2) You spend resources against the Opposing Party trying to get the Ruling Party's favor. If the Opposing Party becomes the Ruling Party, though, you are screwed.
3) You spend resources to dig up the blackmail material on Ruling Party. They become more generous with funding, but become even more likely to sign up the with the aliens.
4) You "replace" them by helping the Opposing Party (who hopefully likes you) to become the new Ruling Party.
4a) Covertly, so that the old Ruling party won't be angry with you in case they come back.
4b) Openly, so that the Opposing Party will be grateful to you. If the old Ruling Party comes back, you are screwed.
5) You completely annihilate the core that holds the party together. The Ruling Party disbands, the Opposing Party becomes the new Ruling Party by default, and the new Opposing Party is created. Bad aftermath rolls may sour relations with all of Council in general.
Optional Complexity: Both parties have Heirs-Apparent who try to become the number one in their respective parties. And Potential Heirs who try take the Heir-Apparent spots.
1) The Boss of the party doesn't like you, but the heir does? Get rid of the Boss.
2) The Boss of the party likes you, but his fall is inevitable and will be succeeded by the Heir who doesn't like you. Discredit the Heir-Apparent, thus ensuring the former Potential Heir (who likes you) will become the future boss.
