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
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 ...