Meaning
The man a woman is cheating with — the other man in an affair, who 'gives bun' to her partner behind his back. Since to 'give bun' is to be unfaithful, the bunna man is the outside lover, the third party in the triangle, and he is a stock figure of dancehall song, gossip, and rueful jokes about infidelity. The word names him plainly and often mockingly, and to be revealed as, or to be given, a bunna man is a familiar drama of Jamaican romantic life.
Examples
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Him find out seh she have a bunna man pon di side.
He found out she had another man on the side.
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Nobody want fi hear seh dem woman have a bunna man.
No man wants to hear that his woman has a secret lover.
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Plenty dancehall tune chat bout bunna man.
Plenty of dancehall songs talk about the outside man.
Origin
From 'bun' (bun/burn) — to 'give bun' means to cheat on a partner; the 'bunna man' is the man doing the cheating with someone else's woman.
How it’s used
A relationship/infidelity term, very common in dancehall. The female equivalent situation is often expressed with 'matey'.