Fiat Seicento suddenly seems as if it has run out of Petrol

My daughter's 2001 Seicento behaved itself for a few weeks after purchase d uring which time I used it as she has a bus pass still. I had a problem ge
tting the key out of the petrol filler cap so I did not bother securing it but just laid it in the filler pipe for a while and shut the panel.
After nearly half emptying the tank it would start to miss and gave all the symptoms of running out of petrol, I have a workshop manual on a dvd and when I saw that the tank pressure was monitored by the engine management co mputer I thought I had tumbled on the answer and sure enough, when I found out how the key came out of the cap and how to tighten it and get the key o ut, it did not occur again for a while. Then it seemed to come back but onl y if the tank was about half full and for a while I topped it up whenever i t got near that. Then a couple of days ago it happened again and the tank was 3/4 full. I topped it up, but still I think the pump was not operating , or so it seemed, I also found that if it started to die - if I switched t he ignition off and on the flow started and away it went again. The engine was quite warm, but not overheated and I got it home by switching the igni tion off and on.
Today - the engine being cold it started OK and I must have driven about 20 miles without a problem, but I am not confident. I asked the Fiat dealer about connecting to the engine management computer but they said this was v ery difficult (and expensive) The petrol pump is in the tank and they do n ot seem to have one and I am not sure this is the problem, anyway, (or the tank filler top, which is about £40.)
It seems it will cost more than I paid for the car to fix this and I am won dering if I can wire a separate petro pump to the ignition switch to bypass the problem.
Has anyone else experienced this and can make any suggestions? I would be v ery glad to hear from anyone.
Thanks and regards George.
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