On my 04 flhtc my gas gauge is not working,is there anything I should check before removing outter fairing?

 

There is a sender unit inside the tank. So check the connector, under the chrome console. Work from there to see if there is any break in electrical discontinuity.

 

Had a bad connection at sender.

 

Well done! That's probably the easiest place to start looking, so its great you've cured it.