Why did Jesus specifically address Peter? Perhaps because Peter had been the one to so boldly assert his fidelity to the Lord, even to the point of death (cf. 26:33,35). This could be Jesus' way of further warning Peter to be on guard, since the weakness of his devotion was starting to manifest itself. "The spirit indeed is willing, but the flesh is weak" - Certainly Jesus did not question the integrity of His apostles for falling asleep, and He would not do such when they forsook Him later that evening. Spiritually, they all had good intentions, but their good intentions would not overcome their physical weaknesses and fear.
"And again a second time, he went away and prayed, saying, 'O My Father, if this cup cannot pass away from Me unless I drink it, Your will be done.' And He came and found them asleep again, for their eyes were heavy" (Matt. 26:42,43). To the Hebrews, the word "cup" was understood to mean "one's assigned portion" (cf. Psa. 11:6; Matt. 20:22,23). On this occasion, it was used as a symbol of Jesus' suffering, which would be culminated at the cross (i.e., His "assigned portion"). When Jesus returned the second time from prayer, He again found His apostles sleeping, "for their eyes were heavy." No doubt they were ashamed of themselves for the stupor which had come upon them, and they knew not how to adequately apologize for it (cf. Mark 14:40).
"So He left them, went away again, and prayed the third time, saying the same words. Then He came to His disciples and said to them, 'Are you still sleeping and resting? Behold, the hour is at hand, and the Son of Man is being betrayed into the hands of sinners" (Matt. 26:44-46). The fact that Jesus prayed the same prayer three times indicates that it is not wrong to use repetition in prayer, but vain repetition is prohibited (cf. Matt. 6:7; Luke 18:7). When Jesus returned from prayer the third time, He again found His apostles sleeping. The time when they could have provided Him with some "moral support" as He agonized over the ordeal facing Him was now gone. The hour of Judas' betrayal was upon Him. They would have to sleep at a later time.
"Rise, let us be going. See, My betrayer is at hand" (Matt. 26:46). It seems likely that Jesus could see a group approaching, perhaps with torches to guide them in the darkness.