FB Messenger keeps running background activity

How did you know that it is running in the background? It is your perception?

With the limited information that you have provided, It is hard to even guess the problems faced by you. 

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Please re-post with detail so that someone else can help.

Here recently I have to charge it more. I really don't use it much so I checked battery usage and it said Messenger background activity used 93% and I am not on it.

If you have Messenger installed it is running all the time, whether you are actively using it or not. Every time a message comes for you it wakes up to save that message.