Influence is not stuck. It has changed in favour of Noblement. The influence began showing the Dukes in Sukua since the first day we expanded into there.
My guess is we can't influence Dukes Influence directly. In the case of HR 7327, we're increasing our relative Influence by decreasing Dynamic Commodities'. You can see this since each gaining faction's Influence increased as a proportion of the value from the day before.
I figure the same thing happened in Sukua, but in reverse with Sukua Noblement. They got a bump in Influence (probably from trading or some delivery missions to them), which resulted in the other factions loosing Influence.
So, Influence is essentially stuck for Dukes, as far as running Missions go.