The main idea of this passage in Matthew is that the Lord has compassion for us. His compassion is shown first by the fact that He went everywhere teaching, giving us the words of eternal life. But also, whenever He saw crowds coming to hear His teaching, He felt sorry for them because of their suffering—both their physical and their spiritual suffering.
By saying the harvest is plentiful, the Lord tells the disciples that it's time for deliverance—the people have suffered long enough. And Matthew's allusion to "sheep without a shepherd" contrasts the spiritual leaders of the day against the Lord.
The disciples' prayer, then, is for the peoples' deliverance from their spiritual oppression, and for faithful spiritual leaders to shine forth—most notably the true Shepherd of our souls, Jesus.
1 Peter 2:25For you were straying like sheep, but have now returned to the Shepherd and Overseer of your souls.
Finally, we note that the readings supplied in Mark and Luke do not in fact come from the same episode as Matthew. Mark's reference to "sheep without a shepherd" comes in his account of the feeding of the five thousand (Mark 6:30–44); Luke's reference to "the harvest is plentiful" comes in his account of the sending out of the 72 (Luke 10:1–12).
The fact these sayings appear in different episodes, however, should not be taken as disagreement or inconsistency between the accounts. Rather, we must remember what John said about this kind of thing: the gospels record what we need to know in order to believe in Him. The Lord said and did so many unwritten things that we should rather be surprised to find so little repetition!
John 20:30–31Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.
John 21:25Now there are also many other things that Jesus did. Were every one of them to be written, I suppose that the world itself could not contain the books that would be written.