MIKE GEORGE says:
The majority of single-trigger over-under guns switch to the second barrel by what’s known as an inertia switch.
In simple terms, this consists of a small weight mounted on the end of a vertical lever.
As the gun moves rapidly backwards rapidly on recoil, the weight tends to stay still, therefore moving the lever in relation to the rest of the gun.
After recoil, a light spring returns it to its original position. It is the movement of this lever that switches the trigger to the second barrel.
If the fine spring that regulates the lever is too stiff then the lever won’t move enough, and so you get no second shot. The usual cure is to regulate the spring so that it exerts just enough pressure on the lever.
This, I believe, is what was done when your gun was fixed. However, there can be other factors.
For instance, a lot of sticky oil in the mechanism can cause insufficient lever movement, and the answer in that case can be nothing more complicated than a squirt of WD40 to wash out the oil.
The problem can also be caused by a shooter changing to lighter ammunition which generates less recoil, and spring regulation is, again, usually the answer.
The problem may become apparent to some people who try the new 21gram loads.
On other occasions – although not in your case – second-barrel failure can be caused by the shooter not releasing the trigger fully between shots.