A mate of mine had borrowed my crybaby not long ago...I got a phone call saying 'sh!t, I think I've broken it! When I click it on, the signal cuts out'
I took it home, put a battery in and suddenly it works.
Basically, I think you need a battery in to complete the circuit, but using a mains adapter stops current being drawn from the battery, so it doesn't go flat.
If you want to do away with the battery, you could try taking a battery clip, solder the red and black wires together, then plug this into the battery clip inside the crybaby to complete the circuit. This is untested, so don't try it if the idea of damaging the crybaby will upset you
If the pedal has a power supply and it's wired correctly,you don't need a battery to complete the circuit...The wah should function with a battery or 9 volt power and with a battery in or not...