There are a few things that can/will cause what you are describing.
You observation of heat soak will be the first issue to cover.
A typical heat soak situation manifests itself on a hot day. This is aggrevated when a portable tank still has Winter blend fuel in it. Is this possible in your case? Does the engine not do this during the colder months?
What is a heat soak? Heat soak normally occurs after an engine (outboard in your case) has been run on plane for a while then shut down. If left for a period of around 20 minutes or so the fuel in the bowls will be the same temp as the engine. The engine has actually increased in overall temperature as well since you shut it down so the fuel is very hot. Add to that the fact that the inside of the cylinder and crankcase is very dry as a result of the on plane run. Now, you try to start the engine. Hard to start and usually will not accelerate once started are the normal observations of heat soak. Your idle and high speed circuits in your carburetor work off of Bernoulli's principal hence the venturi in your carburetor. Once vacuum from the venturi is applied to the lift tubes inside your carburetor the boiling point of the fuel immediately drops and you are now pulling atomized fuel or fuel vapor through the lift tubes instead of solid fuel and air bubbles that make up the needed emulsion. Remember that a liquid bois at a lower temperature when a vacuum is applied or at altitude? Engines will not accelerate on this without engaging the choke or primer liberally. Once the fuel in the bowl has cycled through and been replaced by cooler fuel the engine will accelerate. This may also be the reason why you can do your immediate restart once shut down as the fuel has not gotten hot yet. Moving away from alcohol extended fuels will help some in a situation like this and covering your fuel tanks to get them out of direct sunlight helps a little as well. Go and try this and see if this is what you are experiencing.
There is another issue that may be causing what you describe that is common to older 2 and 3 cylinder loopers that can aggrevate what you are seeing or be the sole cause as well but lets start here first.