Not really as a generic new weapon type, but for new exotics:
Handcannon that goes into special slot. Terrible damage on shielded enemies, but tripled precision damage on unshielded ones and is insanely quick to ready like the plan c
High RoF auto rifle that goes into special slot. Meager damage on unshielded enemies, but will melt shields away with range and accuracy rivaling that of snipers, along with a contained "suppressed" effect that prevents the target from regaining shields. Lasts a short while, but stacks a couple times the longer you sustain fire on an unshielded enemy.
Both make these former primaries fit into the special scenarios archetype that special weapons are supposed to have