I am not a mechanic, but having experienced a very similar issue I did some studying on this forum to find the potential reasons.
I suggest you perform search (right hand side, top window, type in "warm start" or "warm start problem") and you will find lots of information on potential causes and solutions. It is fairly common issue.
I think the way to start the car when warm is to depress gas pedal about 1/3 and crank.
Are you sure the car is overall in tune? Mixture, ignition, spark plugs, etc. - all this is ok? If so, then I can think of a couple of points.
After 10 or 15 seconds of cranking I would stop and screw out a plug to see if it is wet or dry. If wet - the issue may be electric, then you go the usual way about checking electrical components. This may be confirmed with a gasoline smell after this time of cranking. If spark plug is dry, this may indicate fuel delivery issue. The engine is in a very lean conditoin in this state and fuel delivery disturbances are making it even leaner, causing start difficult.
Sometimes it can help to determine if the issue is fuel when you just switch the ignition, but do not crank for 30-40 seconds. If car starts nice then - the fuel pump managed to fill the fuel delivery lines so the fuel is available.
Fuel delivery topic can be related to e.g. insufficient fuel supply to the main filter (there can be many reasons for insufficient fuel supply, this is just an example). This may be checked by testing of the fuel flow at the end of the return fuel line (should be ca0.8L per 15 seconds). If the test fails - you go along the fuel supply chain looking for troubles there.
The main filter may be clogged.
It may also be a trouble with, actually, Thermo Time Switch that is associated with cold start, but is supposed also to help in warm start. Check if you have voltage on CSD when cranking. If not, try to hook a wire to activate "manually" the cold start valve when cranking warm engine - see if it helps. If yes, the issue is with TTS or wiring leading to activate CSD or CSD. Some Members have a manual switch under dash thay added to help inject an extra dose of fuel through CSD when starting warm. CSD may be checked if it works as well if hooking a wire to it does not help.
These are the simplest check ups that com eto my mind, I am sure more experienced Members will advise better. Then you may get into return valves in high pressure fuel lines, etc., but this I would look if the simple tests fail.
I would do the search as I suggested above.
I would also encourage you to visit the Technical Manual.
In the section for Full Members you have a chapter about engine starting aids - you would have to become a Full Member to get there. it is an investment that pays back very, very quickly, actually.
