discretion; consideration; leniency
Refers to intentionally going easy on someone, showing mercy, or adjusting strictness out of consideration. Often used in phrases like 手心を加える (to show leniency) or 手心を見せる (to cut someone some slack).
Since it's your first mistake, I'll go easy on you.
The judges acknowledged his effort and showed some leniency.
Compound of 手 (hand, method) and 心 (heart, mind), suggesting a 'heartfelt hand' or considerate adjustment. The exact historical derivation is uncertain, but the meaning has been stable in modern Japanese.