寝掛け (negake: half-asleep) sleep + to suspend, to hang
Here’s a similar term:
寝惚ける (nebokeru: to be half-asleep)
The kanji 惚 (bo(ke)) also plays a part in 時差惚け (jisaboke: jet lag), which breaks down as time + difference + to grow senile!!! When we’re jet-lagged, it’s as if we’re senile!
In addition, 惚ける (bokeru) means “to become dull-witted, to be out of focus, to fade”—all appropriate for jet lag or for falling asleep.