Is it better to perform the supererogatory prayer (Naflah) that precedes the Dhuhr prayer before the Adhan or after it?

The time for the regular Sunnah prayers begins with the entry of the time for the obligatory prayer they are associated with. For example, the time for the Sunnah prayer of Dhuhr begins with the entry of the time for Dhuhr. It has no relation to the Adhan (call to prayer). So, if you pray before the Adhan but after the time has begun, it is valid. The important thing is the entry of the time for the obligatory prayer.