I read in Malachi 4 that God promises to restore the hearts of children to their fathers. If I claim this promise for myself, may I believe that God will restore my relationship with my children?
It is never appropriate to assume that any particular Biblical promise is addressed to us individually. The vast majority of Biblical promises are made to specific individual or groups (usually Israel) and are not applicable to all believers. For example, the promise in Malachi 4 was not a promise delivered to a Christian today. Malachi 4 is speaking concerning unbelieving Jewish people living in the time of Tribulation.
In Malachi 4, we find the follow prophecy:
Mal. 4:5 “Behold, I am going to send you Elijah the prophet before the coming of the great and terrible day of the LORD.
Mal. 4:6 “He will restore the hearts of the fathers to their children and the hearts of the children to their fathers, so that I will not come and smite the land with a curse.”
The prophecy in Malachi (repeated in Luke 1:17) is a promise speaking to Israel concerning the days of Tribulation. The timing for this promise is said to be "before the great and terrible day of the Lord." This phrase is a reference to the time of Tribulation, a time of retribution and judgment promised for the Jewish people and for the world. Prior to Tribulation, Malachi says the prophet Elijah will return to preach to Israel, and that preaching will result in the hearts of children returning to the fathers and the fathers to the children.
In this context, the phrase "children" is a reference to the descendants of Israel and the phrase "fathers" is a reference to the Jewish patriarchs and prophets. Specifically, Malachi is promising that Elijah's return will result in Israel returning to observing the the Biblical traditions of orthodox Judaism and to following the example of the "fathers of Israel" in their observance of Jewish Law. Elijah's call to return to the Law will lead the nation of Israel to a revival during the first part of Tribulation.
God's purpose in this revival will be to encourage many in Israel to join in worship at the newly reconstructed Jewish temple during Tribulation. You can learn more about this purpose in our Revelation study, specifically in Lesson 11.
God delivers promises to specific people or groups, and so we cannot "claim" a promise in Scripture for ourselves. The Bible never instructs us to engage in such a practice, and there is no mechanism given in Scripture by which we can redirect God's promises to ourselves and apart from His intentions. The concept of "claiming" promises that were originally given to other people is an invention of false teachers, and sadly it has become a part of modern Christian culture.
In reality, God's promises cannot be appropriate for our own desires. Instead, the context of Scripture must guide our interpretation and application of any Scripture, including our application of God's promises. His promises are only applicable to the audience(s) He intended by His word. Only when we find clear intent in Scripture that a promise was intended to apply to all believers universally (as when Christ promised that all believers will receive the Holy Spirit) can we then assume that the promise is ours as well. In the case of Malachi 4, however, the intended audience for God's promise was Israel during Tribulation.
On the other hand, if you believe God has indicated to you personally in some manner that He intends to restore your family or heal a broken relationship, then you can trust in that assurance, for the Lord will do as He has said. If such a work of restoration does happen in your life, you should give God the glory but you should not claim that this work of God was a fulfillment of His promises in Malachi 4.