It is a word more used in the UK than the US. It seems to mean (in US parlance) to whine, but not to be in a position to do anything about the whining's cause, and to persist in whining
So I'm whingeing -- a head cold. If I bend over, my head hurts. Sneezing a lot, and that makes my head hurt, and some other things not to function properly.
And I'm cold. Cold, cold, cold.