Won't stay running

Sounds like you have dirt/debris in your caburetor jets,fuel passages or a diaphgram that is becoming non-flexible.
Check/Clean/Replace your Air Filter, a dirty air filter can make your engine run to rich with fuel.
Note: ALWAYS USE COMPRESSED AIR TO CLEAN YOUR JETS AND PASSAGES, VERY IMPORTANT.
Note:Before you disassemble the carburetor:
Write down on a piece of paper and or take a picture of how the linkage attaches to your carburetor for later reference when you go to reinstall your carburetor.
Mark each piece with a awl, or some kind of sharp instrument that will make an alignment scratch before you disassemble the carburetor into separate pieces.
That way you will know which way it goes back together when you reassemble it.
Sometimes you can get by with priming the carburetor or by using a very small shot of starting fluid and letting it run a few times like that, and it will flush the gunk out of the jets,but most of the time you will need to rebuild the carburetor.
Be sure to check your fuel tank for water and dirt/debris, if there is water/debris then you need to clean your tank.
Check you fuel line/s condition...after a while they will degrade and need replacement.
Check/Clean/Replace your fuel filter if you have one.
When you remove your fuel lines from the carburetor be sure to make a drawing or take a picture of how the lines are connected to the carburetor.
Also a good time to take a picture for later reference for reassembly.
If the mower is over a couple of years old, then I recommend that you buy and install a new carburetor repair kit,because the diaphragm will get hard and that will cause it to be hard to crank.
The diaphgram may look good and flexible, but it can be deceiving and not act as a fuel pump as it should because it has become too hard and will cause hard starting,start and run and shut off, etc.
When you clean your carburetor, I recommend that you use a laquer thinner type cleaner to clean and dissolve the laquer build-up in the float and needle jet passages.
Be sure to remove all plastic and rubber parts before using the laquer thinner because it can dissolve the plastic parts and render them unuseable.
Be sure to use COMPRESSED AIR to blow out all the fuel and air passages.The higher air pressure is needed to blow some of the trash/debris from the fuel or air passages.
Be careful when blowing out the passages, because there are sometimes small rubber type seats in the bottom of some of the passages.
Keep in mind that the float (if you have one) for the carburetor must be level when you go to reassemble the carburetor or follow the instructions you get with the carburetor kit, or you could also ask the parts man that you get your kit from.
When you clean your carburetor and remove the jet screws, you will first need to lightly seat the jet screws.
But before you lightly seat the jet screws count the number of turns it takes to seat the jet screws from their original position.
Be sure to mark the turns down on a piece of paper.
That way when you put the jets back in, you know to lightly seat them first and then turn them back out to their original position before you started.
Once you have your carburetor cleaned/rebuilt that should solve your problem.
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Bud