What does 2 Corinthians 4:1 mean?

In the previous chapter, Paul described the remarkable covenant God has made with those who come to Him through faith in Christ. This new covenant allows human beings to see the glory of God in the person of Jesus Christ. Without Christ, a veil remains between the sin-hardened minds of humans and the ability to see God's glory. Those forgiven for their sin by God's grace, though, are freed from this veil by God's Spirit. As a result, they can look at God's glory with "unveiled faces" and begin to be transformed into the image of Christ as His glory becomes theirs (2 Corinthians 3:17–18).

Paul now returns to defending his ministry to the Corinthians and others. His role is to carry the message of the new covenant to people around the world. Paul insists that he and his co-workers have this ministry, this purpose, by God's mercy. He admits he does not deserve, on his own merits, to carry something as precious and valuable as the gospel of Jesus. God has given Paul this job out of His great mercy.

That's why Paul says that he and his friends do not lose heart. In this context, this means they refuse to quit. They may become "tired in" what they do, but not "tired of" it. They refuse to allow obstacles between them and their mission to discourage them to the point of despair. Despite the difficulties that come with their work, they keep going because God has entrusted it to them. In the following verse, Paul adds that they refuse, also, to cut corners or act without integrity.