Fiesta starting problem.

Hi All,
I bought my wife a UK '93 1.1LX Fiesta with an HCS engine. I've now found out it has an internittent starting problem. Sometimes it starts great first
time and runs without problems. But sometimes it just refuses to start at all! It doesn't even try and start unless the throttle is fully down and even then doesn't quite manage it.
When it finally does run again, after a day or more sometimes if we've flattened the battery trying, it starts and runs and drives perfectly again.
I've checked the plugs and gaps. When the problem is there, there appears to be a good spark. The engine compression is fine. The air filter is new. It has what looks like a carb but is a fuel injection system I'm told. Can anyone give me an idea what to look at or check next?
I don't want to take it to a garage and spend tonnes of money unless I can help it! The fault seems to occur in the mornings only, when the engine is cold.

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