The round black thing is called a biscuit. It spreads the load of the bridge evenly across the centre of the resonator cone. As the bridge vibrates, energised by the strings, the cone moves in and out like the cone of a speaker.
Setting up reso intruments is a pretty specialised craft, whilst I'm certain that the biscuit itself is not likely to be adjustable. If your mandolin is a metal bodied one, and most of the ones I've seen are, there is a joint between the wooden neck and the metal body, which probably needs resetting. This is an even more specialised job, involving the complete removal of the resonator assembly.Once inside, the neck joint could be a simple bolt-on job, like a banjo.
I'd recommend that you search for information on sites that specialise in resonator and slide guitars= which are built the same way.
This one for example :-Reso Guitar hangout
Cheers
Dave