Is it necessary to specify the time and date when making up missed prayers, especially if that is difficult?

When making up a missed prayer, it is not necessary to specify the prayer of a particular day. The Shafi'is do not differ on the fact that this is not obligatory, nor is it even recommended by them. It is sufficient to intend to pray Dhuhr, or Asr, or simply a missed prayer. There is no condition to specify the day, month, or year of the missed prayer.