is mesen emulator allowed for speedruns in NES games

There's no universal agreement for the NES community that it is or isn't. Some leaderboards will accept it, others won't. I personally believe its more accurate than some other emulators accepted and in use right now. But there's no definitive real answer for you.

check the rules for whatever game you want to run. most of them have a list of allowed emulators there.

Mesen is not allowed in many leaderboards for NES games because it uses a run-ahead feature which skips non-input frames in order to reduce lag.

Some communities may allow Mesen, but others don't. Check the rules for the game you want to run and see if they allow Mesen.