expression
without a moment's delay; without even time to
Used adverbially to mean doing something immediately, or being so busy that one cannot spare time for something else. Often followed by a verb.
彼は暇もなく仕事に取りかかった。
He started working without a moment's delay.
忙しくて、昼食をとる暇もなく働いた。
I was so busy that I worked without even time to eat lunch.
暇がない is a clause meaning 'have no time', while 暇もなく is an adverbial phrase meaning 'without even time to' or 'immediately'.
From 暇 (いとま, 'leisure time') + も (inclusive particle) + なく (adverbial form of ない, 'not'). Literally 'without even leisure time'.