What was the driving hope behind the apostles' ministry after Pentecost?
The driving hope was that Israel would repent of crucifying their Messiah, enabling Christ to turn and set up His Kingdom. The apostles knew they could not go beyond Israel's borders until Israel had her King, so their whole momentum centered on Israel's repentance and the promised Kingdom.
Les explains that after Pentecost the twelve were still in Jerusalem, not derelict in duty, but waiting because they could not go beyond the borders of Israel until Israel had her King. Their whole driving momentum was the hope that Israel would yet repent of their sin of having crucified their Messiah, so that Christ could turn and set up His Kingdom. Les connects this expectation to Zechariah 8:20, where God's promise concerning Israel and the Gentiles was still future, because Israel had not yet had the Kingdom.