We won't have proved the diagnosis until we see the mower exhibit the fault, then have it run properly after only making one change, Jaffa. If you change more than one thing at a time, you'll never know what the problem was. (Of course most mechanics doing a job on someone else's mower for money, don't care what the fault was. They just make a broad diagnosis, fix everything it might have been, and as long as the fault has gone, why worry?).
Popping sounds from engines are usually leaky valves, but can be incorrect ignition or camshaft timing. If your description was a bit off, and it was actually a sharp spit rather than a dull pop, the problem would probably be lean mixture, but we've worked together before and I think your description would be more accurate than that. On a Briggs or most other engines with electronic ignition, the only thing that can change the ignition timing is a sheared flywheel key, which only happens when the mower has hit a pretty serious obstruction. Changes in camshaft timing are very common when the engine has been dismantled, but otherwise uncommon on engines with gear-driven camshafts such as your Briggs. If you had an OHC Honda there, with a belt-driven camshaft, incorrect camshaft timing would have been the very next thing to check after the tappet clearances, because it happens to those engines quite a lot.
By the time you started the thread, we knew that you have a side valve Briggs with a gear-driven camshaft and electronic ignition, and you didn't mention having recently mowed a water pipe, so it wasn't going to be ignition timing or valve timing. We also knew that you had very little inlet valve tappet clearance - and an inlet valve that doesn't close properly was likely to cause popping in the inlet pipe. Right away, we had a single most-likely cause.
It wasn't possible to say for sure that the problem was lack of tappet clearance, but it was highly likely. Furthermore the inlet tappet clearance had to be corrected anyway, or you'd end up with a burned inlet valve, if you didn't already have one. That means if you are playing the diagnostic game that I always play, so that you end up knowing what was wrong rather than minimising the repair time, the first thing to do was lap the valves and increase the inlet tappet clearance to the correct amount. That will probably fix the problem. However if it doesn't, you will still have fixed a serious fault, and eliminated the highest-probability cause of the problem - so you would just be left with the same other checks to make that you faced in the first place.
I'd say there is at least an 80% chance that your problem is fixed, but if it's not, please tell us, and we'll move on to some other checks and tests.
It's important to understand the difference between fixing an engine on the first try so you can return it to the customer immediately, versus finding out what was wrong so that the owner will recognise the symptoms next time. Most of us are here to learn, rather than just blindly fix mowers by giving them all a standard tidy-up. Both repair techniques are valid, but they serve slightly different purposes. Lots of Outdoorking members fix most of their mowers by just tidying them up, but there are still times when they want to know what the actual fault was, so they can recognise it next time.
