Is there a special injector or system hooked to an injector, that is used just when you start the car up. This is on a 1991 2.3 My car aways is hesitant to start the first time but always run after the second start.

 

Yes there is a cold start valve that injects fuel into the intake port for 1-2 seconds as you crank on a cold start. Inoperative cold start will make it pretty hard to start the engine.
With it operating the engine should start immediately in a fraction of a second in mild temps. Below freezing may take a tiny bit longer.

 

I agree that the cold start valve is the likely culprit. I would suggest you try to resolve this as soon as you can. Grinding on your starter in order to start the car won't do your starter any favors.either and it can lead to premature wear on the starter as well.

 

yes cold start injector also common to 102 and 103 engines too.
May be clogged up.
Check voltage at injector connecter for a couple of seconds , engine in cold condition

 

Cold start injectors are an easy replacement and act like another fuel injector. Many Bosch FI systems used them. They should be replaced when the regular fuel injectors are being replaced. However, there could be other culprits causing a hard cold start:
Fuel pressure accumulator
Fuel pump
Bad coil & ignition wires