Is it permissible to combine the intention of Salāt al-Tarāwīh (or Tahajjud, or Qiyām al-Layl, or Sunnah prayers) with the intention of making up missed obligatory prayers in a single prayer?

It is not permissible to combine the intention of making up missed prayers with the intention of performing the Tarawih prayer. This is because one of the conditions for combining two acts of worship with a single intention is that they are based on overlap, or that one of them is not intended for its own sake. This does not apply to making up missed prayers and Tarawih, as they are two independent acts of worship, each intended for its own sake. This ruling includes Tahajjud, Qiyam al-Layl, and the regular Sunnah prayers. Whoever has missed obligatory prayers should not engage in supererogatory prayers until they have made up what they owe, with the exception of Witr and similar prayers like the Eid prayers.