In John 1:19-22, John the Baptist is questioned about his identity. The priests and Levites ask him in verse 21 if he was the Prophet. They ask this after he denies being the Messiah and Elijah. So obviously in their minds there was an expected prophet who was neither Elijah nor Messiah.
I can agree that Moses was probably predicting Jesus in Deut. 18, but those questioning John obviously did not believe that. I presume they were aware of Moses's prediction, yet did not think Moses was predicting Messiah (nor any of the already-come prophets, priests, & kings).
What, then, were they expecting of the Prophet if they were expecting someone other than Elijah or Messiah and other than the prophets of old?