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There's no doubt in my mind that this surname is Polish, commonly spelt Borzęcki. Approximate pronunciation Borzhe(n)tskee, the letter -e- is pronounced nasally, though the name is in rare instances pronounced and spelt with a normal -e-.
I can't see the Polish name having a connection with the Hungarian Bor, "wine". I suggest that the Hungarian Borszéki is a different surname, though a Hungarian version of the Romanian Borsec does seem likely.
The meaning of Borzęcki is not clear. It looks like it could derive from a place name.