What is God's sending of Jesus Christ in Acts 3:20 referring to?
It refers to the Second Coming of Christ. God will send Jesus to fulfill the covenant and Kingdom promises made to Israel, as Peter preached. The sending is not a past event but the future return to establish those promised blessings.
Les explains that Acts 3:19-20 connects the times of refreshing from the Lord's presence with the fulfilling of covenant and Kingdom promises. The semicolon shows the thought continues: God would send Jesus Christ, previously preached, for the purpose of fulfilling those promises. That sending is the Second Coming. Israel was aware of Peter's preaching that though they killed Him, God raised Him from the dead, and He can still be their King because He is alive.
And he shall send Jesus Christ, which before was preached unto you: