Learn Russian: У меня есть одно важное знакомство в этой сфере. - I have a certain important acquaintance in this sphere.

This is the indefinite specific use of один. The speaker uses одно to show they have a specific connection in mind, but they are choosing not to name them right now. It functions very much like "a certain" or "this one" in English when hinting at a secret or a specific example.

Is знакомство referring to a person here? I thought it meant the abstract act of meeting someone.

You are right that знакомство often means the abstract concept of "acquaintance" or "meeting". However, in contexts like business or networking, it frequently means "a useful connection" or "a contact". Here, важное знакомство means a specific important person the speaker is connected to.

If I drop одно and just say У меня есть важное знакомство, does the sentence still work?

Yes, it is grammatically correct and means "I have an important connection." However, you lose the nuance of "a certain." Adding одно introduces a sense of intrigue or specificity, implying "I have this one particular connection in mind" rather than just stating a general asset.

Why do we need the word есть here? Don't we sometimes omit it in possession sentences?