There is a conflict of interest somewhere. If the minor station is independant or Federation then this will cause an issue.
If they are Empire aligned then it could be just a local thing where minor factions are competing for the controlling station.
Tbh there is so many little factors that can upset the flow.
If the minor faction has another station try going to that station and starting missions there. You will get influence from there again.
If you have another station that you want the faction to take then try doing all missions in that station. It helps more.
Not all missions will benefit your faction. It depends on what state it has as to what missions help.
Pending states are something to watch out for too. If its not desirable then you have to do things to counter it before it becomes active.
Each faction has an enemy and friendly faction. Find these and trade(friendly) or fail(enemy) their missions.
Each gov. type has a prefered way of doing things. You need to find what this is and use it at the right time.
Other players will spoil things too by doing things that go against your faction, even if they are just there to trade some goods.
Certain 'states' will benefit from a certain type of goods being traded.
Sometimes the missions just take a pause and 15 mins later they just pop again.
It really is a game of chess. Quite interesting. Sometimes you just have to look at the situation and decide what's best for that faction and get on with it and then the missions come back.
I also think failing other factions missions slows down missions given, (it's like they all distrust you) but I'm not 100%.
There is so much more. I will see if I can post some good links for you. I will need to go find them.
brb
Edit: What Power Play is going to do is this, it will decide for us what missions need to be done and present those missions to us.
Right now we have to work out what those missions or activities are.