Meaning of "finesse away"
That's a good list of finds and a good effort. I also looked it up as a phrasal verb, but it turns out looking up just the word "finesse" seems to give an answer:
8.to avoid; circumvent.
Random House Kernerman Webster's College Dictionary2.To handle with a deceptive or evasive strategy.
American Heritage Dictionary2b : evade, skirt
finesse the hard issues
Merriam-Webster Dictionary1.1North American Slyly attempt to avoid blame or censure when dealing with (a situation or problem)
Oxford Living Dictionaries
The first three come from AmE dictionaries, the last one comes from a BrE one, and it makes note that it's a North American use.
Collins English Dictionary (another BrE dictionary) doesn't have a verb definition. All entries have a noun definition for a trick or deception, and in many cases it's specifically used in regard to card games like bridge and whist.
This makes me believe it's particularly a North American thing. This is new to me, I wasn't aware of this meaning of finesse.
I searched Google NGram Viewer to compare British and American results, but no results show. I've also checked the iWeb American corpus search to see what words collocate with finesse, "away" was nowhere to be found. I'd be interested to know whether this is a regional thing in North America or whether it's recognised everywhere there.