It's my understanding that the pump is pressure-regulated. It stops clicking when the bowls are full, resumes running as fuel is being used.
Output voltage at the relay should be battery voltage. If only 6v, check the various points leading to the relay - ignition switch fuse output, engine stop switch input/output, relay input. Yes, could be the relay itself. Bottom line - the pump should be fed battery voltage.
HOWEVER - time to simplify things a tad, Vlad.
I went wayyy back to your original post, you seemed to think it was the fuel pump that was causing your problem.
So, with 12 volts directly applied to the pump, does the bike start and run O.K.? If not, have you checked the fuel filter,and/or the internals of the pump? Have you done a test of the operating capacity of the pump?(pumping fuel into a glass jar).
If the pump works O.K. with direct power, try the same test running off the blue/black wire. If it fails, it's almost certainly because of the low voltage.
Cheers!