Zan in Chinese / Japanese...

First off, 殘心 should only be used in the context of Japanese martial arts. In Chinese, it's a rather sad title (like a broken heart). In Chinese, the first character alone means destroyed, spoiled, ruined, injured, cruel, oppressive, savage, incomplete, or disabled. However, in Japanese, it's remainder, leftover, balance, or lingering.
The second character means heart, mind, soul, or essence in both languages.

殘心 is one of the five spirits of the warrior (budo) and is often used as a Japanese martial arts tenet. Under that context, places such as the Budo Dojo define it this way: The spirit of zanshin is the state of the remaining or lingering spirit. It is often described as a sustained and heightened state of awareness and mental follow-through. However, true zanshin is a state of focus or concentration before, during, and after the execution of a technique, where a link or connection between uke and nage is preserved. Zanshin is the state of mind that allows us to stay spiritually connected, not only to a single attacker but to multiple attackers and even an entire context; a space, a time, an event.

In modern Japan (and Simplified Chinese), they use a different version of the first character, as seen to the right. Click on this character to the right instead of the button above if you want this modern Japanese version of lingering mind / zanshin.

残心 is a Japanese Kanji word meaning: continued alertness; unrelaxed alertness; remaining on one's guard; lingering mind, and being prepared for a counterstrike. This context is used in martial arts, which is probably why you are looking up this word.

In archery and golf, it can be the follow-through.

In the context of love and relationships, it can be lingering affection, attachment, regret, regrets, or reluctance.