Hi - As shown in the image/diagram below
there are flexible tubes that transfer ink from the cartridges to the printhead, in Brother MFC printers. Look inside your printer, and see if the black tube (or any of the tubes, if you cannot tell which one is black) is odd looking in any way or broken, or obviously leaking.
Also, check around inside the printer generally for fluff, dust, leaking ink, paper scraps, etc - if found, clean them out (by wiping them out, or with a vacuum rather than by blowing, and with the printer unplugged of course.)
Also, check under the black cartridge for any sign of dried ink or dirt accumulation or clogging or the like - if necessary, compare it to another (working) cartridge's bay/socket.
(By the way, how old is this printer? years in service, I mean.)
Let me know what you find.