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Nicolas Cage had something he clearly wanted to get off his chest when he hit up a karaoke bar ... because his rendition of "Purple Rain" was ... emotional.

Nic was at a joint Sunday in L.A.'s Koreatown when he tried his hand and vocal cords on the classic Prince song, and we're guessing the performance was fueled by his desire to annul his recent marriage of 4 days.

As we've reported ... Nic got sick and tired of his new wife, Erika Koike and filed for an annulment. He says they were wasted when they tied the knot in Vegas, and acted on impulse. There were signs it wasn't going to end well ... they got into a heated argument before walking down the aisle.

Erika doesn't see it that way ... she says the marriage was legit, she claims Nic asked her to come back to him, and she wants spousal support.

As for "Purple Rain" ... Prince once broke it down like this, "Purple rain pertains to the end of the world and being with the one you love and letting your faith/God guide you through the purple rain.”