B767 Captain reported TE EDGE FLAP ASYM master caution message on approach to their destination resulting in the flight crew executing a go-around and setting up for an approach at the airport. The aircraft landed safely at their destination airport.

Narrative

The flight to ZZZ was mostly normal other than we had discussed with dispatch a diversion strategy to ZZZ1 due to degrading weather in ZZZ. As we started descent we noted that the weather was just slightly above minimums for the approach; 1000 meters RVR and 200 feet broken. The approach required 800 RVR and 200 feet. We elected to make one attempt before heading to ZZZ1 which was nearly 3 hours away. This alternate was chosen because ZZZ is fairly remote and the company felt that ZZZ1 would be able to provide the best handling services. We were vectored onto the approach all was normal until roughly 30 seconds after flaps 30 was selected; TE EDGE FLAP ASYM master caution message displayed. I elected to discontinue the approach and run the checklist to resolve the issue to the best of our abilities. We conducted a high altitude missed approach and quickly realized we were unable to retract the flaps. The gear horn was sounding and was extremely distracting; there is no way to silence this other than extending the gear; I elected to do so as we were in non mountainous terrain and at a low enough weight that performance for now was assured. The pilot monitoring in the 1st observers seat ran the TE FLAP ASYM non normal checklist while the first officer manually flew the aircraft. I felt that due to aircraft configuration that the autopilot may not be the best choice. I managed the situation requested a vector from ATC and trouble shot with the pilot observing. I quickly realized that due to increased approach speed and the checklist being unsure of what flaps were actually extended that there were very few alternate airport options if we were unable to land at ZZZ. Continuing to ZZZ1 with the flaps at 30 and gear down was not an option. We likely had an hour and a half of fuel at 165 kts maximum airspeed due to configuration giving us an absolute maximum range of 247.5 miles. We also required more than 6000 feet of runway. We started the approach to ZZZ one more time; at minimums I saw approach lights we continued 50 feet and just barely saw the runway threshold lights. We were able to land but it was possibly the lowest category 1 approach I have ever conducted. The aircraft was limited to category 1 and that was the only available approach at ZZZ. We noted that the weather looked improved about 1 hour after landing so we may have been able to hold and wait it out. Maintenance gave us no answer as to how he resolved the flap issue he seemed unsure of why it happened. We reluctantly continued on to ZZZ2 which we had major concerns about the issue repeating itself now at a high altitude airport but after running numbers with the fleet captain I was assured that at the low weight I was arriving into ZZZ2 with I would have the performance to land or divert to another low altitude airport. Enroute to ZZZ2 we learned that TE FLAP ASYM was a repeat write up from maintenance. Sadly repeat maintenance items are not uncommon it seems the norm to do the bare minimum to resolve most maintenance issues here. I cannot emphasize enough how few options we would have had if we were unable to land at ZZZ. Also this specific 2 leg day was incredibly fatiguing; there are no rest facilities on the aircraft. There is no where to sleep; augmented crews on late night departures with extended duty days are not safe.Cause: Poor maintenance; poor planning; remoteness of ZZZ; lack of good alternates.Suggestions: We need a closer alternate than ZZZ1 although it is useful to the customer although given the aircraft state there is no way we could have made it there.