The phrase "to influence the judge" means trying to persuade or convince the judge to make a decision that is favorable to you or your side in a legal case. Full definition
This case highlights that perceived «bad behaviour» may influence a judge who is being asked to exercise his discretion in your favour. (idiproject.com)
I'd also argue that the culture of blogging might properly influence a judge's assessment of whether an author could assert that a posting represented a reproduction of a work. (slaw.ca)
Parent behavior during a custody battle can influence the judge as he decides which residential placement meets that criteria. (info.legalzoom.com)