Does the intention of making up a prayer affect it if the prayer is not, in reality, a make-up prayer, and is a prayer considered a make-up prayer simply by its time expiring, even if it is a repetition?

If a Muslim prays believing that the time is still current, so he intends performance (adaa'), and then it becomes clear that the time has expired; or if he believes that the time has expired, so he intends make-up (qadaa'), and then it becomes clear that the time is still current, his prayer is valid, as long as he did not do so intentionally. This is by agreement of the madhahib (schools of thought). Many jurists hold the view that the intention of make-up (qadaa') or performance (adaa') is not strictly necessary, as the time itself determines this. However, it is advisable to intend performance or make-up to avoid scholarly disagreement.