Nope.
It's a big problem with mission wrinkles... they add credits, but not influence.
The only influence comes from the bounties you earn, when you hand them in (unless collected via IF)... so it is viable to deliberately take a bunch of missions to generate tails and go hang in the jurisdication you want to support in a combat ship to earn bounties for that faction. I find stacking Salvage missions to be good for this; with 10-20 under your belt, you have a 50/50 chance of getting a tail once you've collected the salvage. Because that's no more than 40-80 pieces of cargo, it's very easy to field a combat-ready shipand do these simultaneously.