It's often the case you'll want a favour from a high-ranking public official or from someone who works in an area you want to exploit for profit. Thankfully these people are often open to bribery and corruption.
Each corrupt official has a cost and a reliability/integrity (R/I) score.
The cost is (fairly obviously) the amount you must pay them each round so that they'll work for you. Bribery only works if you keep on paying. If you cut your payment for two rounds or more then the corrupt official must roll a 1d6 R/I check. If he succeeds (lower than his R/I score) then he won't work for you again.
The other option for gaining someone's loyalty (or at least their compliance) is blackmail. This relies only on the ingenuity of the mafia. If they can come up with a plausible plan, the corrupt official rolls an integrity check. If the check fails then the plan works and the corrupt official is under blackmail. An example of a plausible plan is: "We offer the judge the free use of one of our whorehouses. When he's "busy" upstairs we bring in a representative from the Tattler (who're on our payroll). He takes pictures and we threaten to go to press unless the judge acts for us." The plan must be acted out in the B-round (with the obvious censorship!) when the R/I check will be made and the audience will decide on the plausibility of the plan.
Each corrupt official may be bribed by only one mafia Family at a time. When you ask a corrupt official for a favour, make a reliability check (1d6). If you roll their reliability or less then you can consider the favour done.
Note that you can also try bribing Civilians who work in the right area (e.g. lawyers, judges, newspaper tycoons).