mediation; intercession; go-between
The main modern use: arranging an introduction, speaking on someone's behalf, or using one's connections to help something happen. In politics or business it can imply questionable influence or intervention.
Through an acquaintance's help, I got a chance to interview.
The circumstances of the contract were investigated on suspicion that a politician had intervened.
influential person; person whose word carries weight
Less common as a standalone noun than the mediation sense. Refers to someone with enough influence or connections to make arrangements, close to 顔利き.
My grandfather was a person of some influence locally and had connections at city hall as well.
way of speaking; eloquence; eloquent person
Archaic speech-related use. The original sense was a person's manner of speaking, and it could also refer to eloquence or an eloquent speaker; modern Japanese normally uses other words for these meanings.
In old writing, 口利き can refer to one's manner of speaking or skill in speech.
I asked the department manager to intercede on my behalf.
I found a job through an acquaintance's connections.
He always acts as a go-between for others.
The lawmaker resigned over allegations of improper mediation.