Starts
Mr. Flyguy.
Five pumps is a bit over kill unless it's 10 below outside. If your last fiights were getting "worse", you may have other issues to address. Most Lycs use the left mag for starting, although some are equiped with dual impluse. It's unllikely that one is even dead and virtually impossible to loose both, if dual equiped.
You described the"last flights", that the engine wasn't running normally, then indeed there are other issues. Starting with the easiest, check the spark plugs for fouling, if they are wet, there are even more issues. Plug wires if aged will weaken spark. The magnetos themselves can have weakening condensors, worn or pitted points, or improper gap.
On the issue of priming, any 2 or 4 cycle piston engine requires more fuel for starting than when running. This is due to temperatures of the engine itself/ As OAT drops, so is more fuel required. On a typical Floriday day, at 80 to 90 deg, one shot is enough, but when I lived in the windy city, at -10, five or so was not abnormal. In no case should you use the pump for starting. It will invite a fire.
A cold engine requiring fuel, if not enough of it is used for starting, will constitute a lean mixture. Lean mixture will back fire. An air filter with raw fuel in it is ripe for fires. Lycs are outfited with primers for that reason, as are Contis, Franklins and other carburated engines.
It was not unusual for early Cubs and similar hand propping ACs to "flood" the carb with just enough fuel so that vapor made its way to the cylinder. Hand propping would also allow this due to the "slow" movement of the engine. These engine had slower timing than most Lycs have, 18 to 22 deg vs 25 for most Lycs. and less likely to back fire, though they occasionally did.
Having witnessed starting a lean engine (cold) go up in flames on more than one occasion, it's best that primers rather than accelerator pumps be used for cold starts, Always remember that a cold engine is lean. and lean is ripe for backfire. More importantly, correct your problem.
T88