2. Thy kingdom come. This petition has plainly a reference to the doctrine which Christ preached at this time, which John Baptist had preached before, and which he afterwards sent his apostles out to preach--the kingdom of heaven is at hand. The kingdom of your Father who is in heaven, the kingdom of the Messiah, this is at hand, pray that it may come. Note, We should turn the word we hear into prayer, our hearts should echo to it; does Christ promise, surely I come quickly? our hearts should answer, Even so, come. Ministers should pray over the word: when they preach, the kingdom of God is at hand, they should pray, Father, thy kingdom come. What God has promised we must pray for; for promises are given, not to supersede, but to quicken and encourage prayer; and when the accomplishment of a promise is near and at the door, when the kingdom of heaven is at hand, we should then pray for it the more earnestly; thy kingdom come; as Daniel set his face to pray for the deliverance of Israel, when he understood that the time of it was at hand, Da 9:2. See Lu 19:11. It was the Jews' daily prayer to God, Let him make his kingdom reign, let his redemption flourish, and let his Messiah come and deliver his people. Dr. Whitby, ex Vitringa. "Let thy kingdom come, let the gospel be preached to all and embraced by all; let all be brought to subscribe to the record God has given in his word concerning his Son, and to embrace him as their Saviour and Sovereign. Let the bounds of the gospel-church be enlarged, the kingdom of the world be made Christ's kingdom, and all men become subjects to it, and live as becomes their character."
Source: Matthew Henry's Commentary on the Whole Bible Volume V (Matthew to John)
Verse 10. Thy kingdom come. The word kingdom here means reign. Mt 3:2. The petition is the expression of a wish that God may reign everywhere; that his laws may be obeyed; and especially that the gospel of Christ may be advanced everywhere, till the world shall be filled with his glory.
Source: Barnes' New Testament Notes