In the PollingPlace FC, I calculate the POLLINGID field as the VotingPrecinct FC's NAME field for which the Polling Place point resides. This is a 1-M relate so you can have more than one record with the same POLLINGID. You'll want to make sure the POLLINGID in the PollingPlace FC matches exactly with the VotingPrecinct FC's NAME field or the next step will not work. After you have calculated the PollingPlace FC you can load the records into the PrecinctPollingPlace table via ArcCatalog. Then in ArcMap, you can join this table with the VotingPrecinct FC via the matching join fields. Now you can calculate the PRECINCTID in the PrecinctPollingPlace table.
Clear as mud? Good Luck!