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to complain fretfully: whine
"I shall have one pint of beer less every time I'm in the pub, and I might occasionally whinge about the rises in the cost of living outstripping my wages." blog post on A Dull Day at Work, April 2, 2011
About the word:
Whinge (short "i" and terminal "j" sound) predates the long "i" whine. The Old English ancestor of whinge meant "to moan."