A TRACON Controller reported an aircraft read back the wrong altitude contained in the approach clearance and descended below the minimum vectoring altitude.

Narrative

I was training a new trainee. The weather conditions were poor but traffic was light. Aircraft X was vectored for the ILS Runway 3 approach over a 4100 ft. MVA. My trainee issued the approach clearance and stated 'Maintain 4500 until established' but the pilot read back 'maintain 3500 until established'. Neither of us heard it; and I specifically remember at the time we had been discussing legal turn on's for the ILS in bad weather; and I was pleased that the trainee had achieved exactly 3 miles from the FAF. I think having that conversation while working traffic was a distraction. As my trainee instructed Aircraft X to contact Tower I noticed his altitude was 3800 ft; and the crossing altitude at the FAF was 3900 ft. It's not unusual for altitudes to fluctuate by 100 feet so I told my trainee to get ready to issue a low altitude alert if he continued down to 3700 ft. When I saw 37 on the data tag I said 'low altitude alert' with the intent of getting my trainee to call local and instruct them to issue it. I wasn't clear enough and this resulted in him issuing a low altitude alert on our frequency; but he had already switched Aircraft X. He then called local and had them do it. I'm told that by the time the alert was issued; Aircraft X was past the FAF and opted to go around. Suggestion: I need to be more vigilant about allowing training related discussion to distract from actual live traffic being worked by my trainee. The discussion about legal turn on's in bad weather could have waited for the debrief. I think if I were only focusing on listening to the traffic I would have caught the bad read back.