Hi Toni, welcome to Outdoorking. Finding problems can be a complicated procedure, or you can be lucky and find they have a simple and more or less obvious cause. Unfortunately it requires a Sherlock Holmes process unless you just take a scatter-gun approach and give the machine a general tune-up. At Outdoorking we usually attempt to play Sherlock, because it is much more fun.

As mark said, it will probably be worthwhile downloading the workshop manual for your engine. You can get it here:
http://www.toro.com/en-us/parts/Pag...Lm2IVhOJIHhZ9wIClCEiu122kfJiC2/A08JVarky

Click on the "Manuals" tab. Then move down the page to the manual identified as "Service Manual (English) 492 0260".

The Sherlock route to problem solving involves paying careful attention to the symptoms, and pursuing only the potential problems that are consistent with the symptoms.

In this case, you have reported that the engine will not restart when it is warm. However the particular circumstances of your terraced garden suggest that it may also be that the mower will run for a fairly specific amount of time, then it stops and will not restart for another fairly specific amount of time. When that happens with a two stroke engine, whether on a mower, a line trimmer, a chainsaw or whatever, there is one simple fault that is often at the heart of the problem. Your engine can only get fuel out of the fuel tank if air is able to get into the fuel tank to replace it. This happens through a plastic (imitation rubber) vent in the fuel tank cap. Often with time, dried-up 2 stroke mix in this vent oxidises and gums up the vent, so no air can get in. You can test for this problem simply by loosening the cap half a turn and trying to mow the lawn. If the mower then works perfectly, you need to clean out the tank cap vent.

Please try this possible solution then come back and tell us whether it worked. If it doesn't work, we'll return to Sherlock thinking mode. If it does work, we'll close this thread. If it works and you don't come back and tell us, we may never speak to you again. I can promise I won't, for one.