737-700 Captain reported a trailing edge flap problem resulted in a go around to perform the QRH procedures. The flight crew then requested priority handling and continued the approach at destination airport.

Narrative

On approach in to ZZZ; whilst conducting OE on a new hire First Officer (FO) we had a trailing edge flap malfunction. We were flying a 737-700; which has a flaps 30 speed of 165 kts. We were told to maintain 180 kts. to FAF; which occurred at about 1;500 ft. AGL. We therefore had approximately 500 ft. to slow the aircraft and select flaps 30 to meet stabilized approach criteria.The Pilot Flying (PF) requested gear down and flaps 15 approaching glide slope intercept. As I was doing OE; my focus was on watching the FO slow down to 165 kts. so we could select flap 30. I did not notice that when selected to 15; the flaps had stopped at about 8 degrees. When the FO requested flaps 30 and landing check; I ran the checklist and saw that the flaps were not at 30. By this time we were at 1;100 ft. AGL. I instructed the FO to initiate a missed approach and informed Tower that we were going around due to a system malfunction. The missed approach was well flown by the FO; and I felt comfortable having him continue to fly while I worked the problem; and I set the speed at 200 kts.; so not to over speed or stress the flaps. I called back and told the Flight Attendants that we had a malfunction and would need a few minutes before returning to land and that I would call back when I had time and brief them more comprehensively. I made a PA to the passengers that we had an abnormal indication and we would return to land as soon as we had finished our checklists; and that there was no reason to be concerned. I then sent an ACARS message to dispatch alerting them that were would request priority handling due to a flap issue.We completed the QRH procedures; then requested priority handling with ATC; as we would be landing in a non-normal configuration. I called back to the Flight Attendants and briefed them using the TEST method and informed them that we would land normally and that they could expect a normal taxi to the gate. We followed the QRH and selected flaps 15 using the ALTN system; and landed normally. We informed ATC on the ground that we did not require further assistance; but a truck followed us to the gate per their protocol.At the gate we met with maintenance personnel and spoke with the Chief Pilot.We both felt comfortable and able to continue with our scheduled flying; so we did.