What was Paul's "thorn in the flesh"?
The question is a good one. Paul is very vague here about what he has in mind and doesn't mention it anywhere else. Flesh in the NT is used usually either as our physical body or as our fallen human nature. Most interpreters to Paul's statement in 2 Corinthians 12.7 have chosen the first. Paul is referring to a physical impediment. This would be my position. What this might, however, is a matter of speculation. It could be a speech impediment or a faulty limb, or something quite different.
Another possibility is that Paul goes on to speak of the persecution he receives because of his witness to Christ. His topic, however, is not persecution, but his weakness in contrast to his proud opponents at Corinth, who claimed to be so possessed by the Spirit of God that they were beyond such weaknesses. Paul's response to them is that my weakness allows me to know the power of God in my life versus my own strength. The persecutions (v. 10) are an example of weakness, but he first cites them as a parallel to the weaknesses he has been discussing (.10a), not as explanation of them.
That flesh is his fallen human nature does not fit with his theology of the transforming grace of God that God does indeed make available to those who seek it through faith in Christ. e.g. Rom. 6.14; Phil. 3.8-9.
The identity of Paul's "thorn in the flesh" (2 Cor 12:7) has elicited much scholarly ink and no overall consensus. In general, three interpretations have been offered. First, it refers to some psychological or spiritual anxiety suffered by Paul. For instance, a long line of commentators have argued that he was afflicted with sexual (carnal) temptations (often based on the Latin translation stimulus carnis). Second, the "thorn" refers to some physical malady, such as headaches, fevers, or problems with his eyes (see Gal 4:13-14). This line of interpretation is widely accepted today. Third, it refers to those figures who opposed Paul in his ministry of proclaiming the gospel and founding churches (e.g., the “superapostles” named in 2 Cor 11:5 and 12:11). In my opinion, this third line of interpretation, stemming as far back as St. John Chrysostom, is the most probable. The word “thorn” is used metaphorically in the Old Testament to describe Israel’s enemies (Num 33:55; Ezek 24:28). It is striking that, after mentioning the “thorn,” Paul then writes of a "messenger" or "angel of Satan" who continually “beats” and buffets him. This echoes his statement in 2 Cor 11:14-15 in which he describes the intruding missionaries as Satan’s ministers.