There is a lot of information on how the pilot should ask the ATC to repeat a statement, but is there a standard way for the ATC to ask the pilot to repeat a readback, or any new information that the pilot provides?
Definition: Used to request a repeat of the last transmission. Usually specifies a transmission or portion thereof not understood or received; e.g., "Say again all after ABRAM VOR."
Notice this glossary is specifically "pilot/controller" they use the same terminology. There is no separate "controller-only" and "pilot-only" glossary of ATC terminology.
Controllers routinely monitor multiple frequencies and shout lines. Typically, the air-to-ground communications (pilots) is directly in a controllers ear, while landline, or shout-line communications (other controllers), are in the overhead speaker. When a controller answers the landline, the communication is reversed. The landline (controller) is now in the controller's earpiece, while the air-to-ground communication (pilot) is in the loud speaker.
We give the other controller our undivided attention to ensure proper coordination. Remember, we're often coordinating lots of numbers. We hear the pilot calling, however we typically don't retain the details clearly. Hence, we'll say we were on a landline, please say again.