rman cloning rman-20207 error
No, it does not remove backups. They are just missing from the controlfile, and you can catalog them.
When the T database was refreshed yesterday from P, a new incarnation of T started. But your UNTIL TIME clause refers to a point of time before that happened, when the previous incarnation was active.
After restoring the controlfile on C, mount the database, then you can use the following command to view the incarnations:

RMAN> list incarnation;

You will also see the times when they started. Revert to the previous incarnation, e.g:

RMAN> reset database to incarnation 1;

Catalog the backups:

RMAN> catalog start with '/path/to/the/backups';

Then continue with restore and recover as you normally would.