Is Special Ministry Required to Break a Medical Condition That Runs in a Family?

No.

The same Word that heals everyone else will heal someone with a family history of a certain disease.  There is no special secret.  There is no biblical record that Jesus or any of His followers ministered in any different way to someone with a family history of an illness, including the breaking of any “generational curse.”

Some extra ministry might be required to convince the person that he does not have to follow in his dead ancestors’ footsteps, but that kind of ministry is just more sharing of the Word, not some spiritually different ministry requiring obscure special spiritual knowledge.