Hi Grogoreo, it's a long time since I spent time in Liverpool's folk clubs and sessions but I know without going too far outside the centre, there is a plethora of music nights of all description just waiting for you to turn up.(that's if you can find your way through the biggest building site in the north). Just a couple to get you started:

Everyman Folk Club, Hope street (in between the two Cathedrals)on a Tuesday night.

Pogue Mahone Irish Session, Seel Street.(Leave the car at home) ;)on Wednesdays.

Enjoy!!