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?
Yes, and it is exactly the same: Say again.
Station calling Georgetown Ground say again your callsign
Lufthansa 123 say again speed
Scandinavian 123 say again all after 'good morning'
United 345 say again
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.