work hard, belabored, toil
勩 centers on exhausting effort: working so hard that one becomes worn out or belabored. The idea of toil is the natural result of pushing oneself to the limit.
勩 combines 貰 (a component associated with effort or exertion) and 力 (strength/power), suggesting the idea of using strength to the point of fatigue. The exact historical development is uncertain.
The left side 貰 looks like a heavy load being carried, and the right side 力 is strength. Picture someone straining under a burden until they are completely worn out.
For エイ, imagine an exhausted worker letting out a long sigh: 'Eeei...' The drawn-out 'ei' sound cues エイ while the person collapses from overwork.