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The specifications for your model of camera were the first thing I checked. It does say that it is compatible with SDHC cards.
And 4 GB is not a very large size for an SDHC card, so it isn't a problem of size limits.
I would say, that if the problem shows up with only one card, you may have a problem card.
If the problem shows up with more than one card, you may well have a defective memory card slot in the camera. (This especially if the cards work in other card readers or card reading devices.)
A footnote in the camera's specifications says 'see the FujiFilm website to check memory card compatibility', but I doubt that means anything much.
Try removing the batteries, and leaving the camera on a (dry, non-humid, non-dusty) shelf for overnight. Then put the batteries back in and try the camera again.
How old is the camera? in months or years since you bought it.
Has the camera been dropped or otherwise subjected to mechanical shock(s) in its lifetime?
Has the camera been operated in or subjected to, temperatures below 32 degrees F (freezing point) or above about 100 degrees F ?
Has it ever been subjected to a humidity higher than 85 % or gotten wet ?
Has the camera been operated in a gritty or sandy environment, such as a beach, a seaside, a construction site, or the like?