Can an employee influence their supervisor's decisions more than they think they can?

Yes! Especially if their relationship is not a neutral one.

For example, if the relationship is friendship; or if the relationship is bad, it would likely, directly affect a supervisor's decisions. If the relationship is kept neutral; there would be less of a favorable or unfavorable influence.

Good supervisors strive to focus on performance to make decisions rather than personal feelings. But supervisors are only human.