If your Esprit has a hard time starting after it's been operated for a while, you probably have a fuel system problem.
Check the float heights and weights in your carburetors. Spec is 15mm height and 8gr. You may have a bad float which floods the engine making starting difficult. Letting the car sit allows the gas to level off in the float bowls allowing you to start the car. Does your oil have a strong gas smell?
Do a compression check. If your pistons are crowning (famous on pre-86 Turbos) the compression may be insufficient unless the engine is cold and everything can contract. You can purchase an inexpensive Compression Tester Tool that will check the compression of your engine (~$35.00US).
Replace the fuel filter (blockage or partial one may be present), check fuel pump operation and fuel pressure regulator. Make sure all vacuum lines are tight and in good order.