I suggest going about it this way. First, adjust the main mixture screw to halfway between too rich and too lean, with the engine running at fairly high speed. Check that this setting is valid under load by doing some mowing or running the mower uphill. Then forget about the mainjet for now, it is probably correctly set. Do you have an idle mixture screw? If so, adjust it the same way: with engine at idle speed, turn the screw slowly clockwise until the engine slows down (too lean) then turn it anticlockwise until it starts to run poorly - slows a bit and runs roughly (too rich). Then put it halfway in between.
If there is no idle mixture adjustment, the first thing to resolve, is what is the manufacturer's recommended idle speed? For a B&S engine it is 1750 rpm, which is hardly an idle at all. B&S object to their engines being run slower than that. The reason they give, is that if someone slows the engine to idle when it has been running under load, it will suffer from heat-damage: the flywheel fan is almost ineffective below 1750 rpm, but the cylinder head and valves will be very hot when the engine has been running under load. If you suddenly remove the cooling provisions, you get a "heat soak-back" effect: temperatures inside the engine will rise to a damaging extent, before they begin to fall.
You can get an idea whether your engine is rich or lean at idle by removing the air cleaner, which will make it very slightly leaner, then turn off the petrol feed from the tank: does it gradually run better as the fuel level in the float-bowl drops, thus making it leaner? Then slightly obstruct the carburetor air intake with an object. (Do not do this with your hand, that is a very bad habit: engines running lean often spit burning petrol out of the air intake.) Obstructing the air intake makes the mixture richer. Does it run better or worse? After you have tried both richer and leaner, you should know whether you need to make an idle mixture adjustment, and in which direction.
Now for the hair-raising part. If you decide that Tecumseh got it wrong, or somebody has loused it up since it was manufactured, you can make a permanent adjustment. If it is close to right and you just want to tweak it a bit, adjust the float level lower to make it leaner, or higher to make it richer. You do this by bending the float-arm. Before you do this, be sure to measure the float level as it is now. That is the distance from the top of the float to the underside of the float bowl lid. You measure it by turning the lid upside down, so the weight of the float holds the needle in the closed position. If everything you try makes things worse, you can always put it back as you found it. Remember, an adjustment of a sixteenth of an inch is significant. When you get the idle right, you will have to readjust the main mixture screw, since you will have altered the float level and this changes everything.
If the situation is badly wrong at present, rather than just slightly wrong, stronger measures are required. I once had a little OHV Villiers that ran appallingly rich. I lowered the float level a quarter of an inch, and it was better but still rich. I was too lazy to fix it properly, since all it did was power a duck-plucker that I never used anyway, so I left it at that. I noticed, though, what had caused the problem: some clown had run a drill-bit through the main jet, enlarging it considerably. The way I customarily fix that on old cars that have been got at (it isn't an unusual situation, with old engines) is I solder up the brass jet and then re-drill the hole through the solder, using a tiny high-speed drill or a pin-vice. You need some special drills to do this, usually in size increments of .001" (called jet drills). A set of these drills is very expensive, so all I ever did was buy two or three sizes around the size I needed, and probably lose them before the next time I needed them. If you drill the hole a bit too big, you just solder it up and start again. Do not expect to be able to use ordinary drills for this job: the steps between sizes are much, much too big.
