The church suffering is a category particular to Roman Catholic theology regarding its beliefs about people in purgatory. Perhaps the term the church suffering is best understood in contrast to two other terms: the church militant, which is the church alive on earth, currently involved in spiritual warfare; and the the church triumphant, which is the church currently in heaven, comprised of those who have died in Christ.
Although “the church suffering” might sound similar to what evangelicals mean when they speak of “the persecuted church,” “the church suffering” is something completely different. The church militant is on earth and would include “the persecuted church” and any other Christians who suffer in any way on earth. The church triumphant is in heaven. The church suffering is somewhere in between—purgatory.
In Roman Catholic theology, purgatory is actually part of heaven—the vestibule of heaven, as it were. Those who died in a state of grace but had unconfessed venial sins go to purgatory to suffer and pay the penalty for those sins. According to this theological system, they are part of the church, and they will eventually become part of “the church triumphant,” but, for now, while they are being purged of their sins, they are part of “the church suffering.”
Of course, we would object to this category of believers. All sins are mortal, and we have no ability to pay for any of our sins, either in this life or in the next. Salvation is a gracious gift from God through Christ. For the Christian, to be absent from the body (death) is to be present with the Lord (2 Corinthians 5:8). There is no purgatory, as Jesus Christ has paid for our sins in full. The idea of the church suffering in purgatory is not supported in the Bible.