Actually, head is usually the favourite morsel - when the males bring in food, rarely does it still have its head, and its not for aerodynamic reasons
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When courting, if the male brings "gifts" with heads, he's definitely smitten (or seriously not letting this one get away). One year, Trey's Mom was courted by a male that laid out about 20 different species of birds (there were some repeats of species too) on all the ledges all around the building at nest box height. I had a wicked and wonderful time peering through the louvres trying to identify them. After a couple of days, she accepted his suit and all the "gifts" disappeared - and no, she didn't eat them ...
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