Your camera is possibly still under Fujifilm's one year warranty, the only safe thing to try before sending the camera for service is to use canned compressed air to blow out the area between the lens barrel and camera body since even a small amount of sand or grit can cause this problem.
You can use the information at the link below to contact Fujifilm directly for free repair or replacement.
If for whatever reason you are unable to use the warranty service, please see the procedures below.
Put the camera switch in the Off position. Place it on the back with the lens facing up and take a look at the spacing between the lens and the lens housing. If you notice that the gap is not even all the way around the lens, the problem should be easy to fix. This type of a problem usually occurs if the camera was accidentally turned on while the lens is restricted.
Next, apply gentle pressure down the lens on the side where the gap is the biggest. You should hear a "click" as it pops back into place. Try powering the camera on.
If the lens doesn't extend at all or it extends, and then retracts again, do the following. Turn the camera off. Take the camera in one hand and with the other gently take one part of the lens and gently move it round in a circular movement. Do so with both sections of the lens. You will hear a "click" as it pops back in place. Power the camera on.
Next, try to pull and twist on the largest ring of the lens while turning the camera on. Listen for a "click". If at first the focus seems to be off, turn the camera on and off and take lots of pictures, close ups and distance. Focus should slowly start improving.
If the steps above do not work or there appears to not be enough power, depending on how long you have been using the rechargable battery, it may be time for a new one since as digital cameras begin to age they become more sensitive to the power level.
Since the lens does not make it to the correct sensor point due to it being forced in, as noted above, some physical manipulation is necessary. A lens position error will halt the camera start up process. Lens errors are a common problem. With newer cameras I am usually hesitant to suggest these options since they include some more extreme procedures that may cause further damage if not done correctly, but they may save you an expensive repair bill or having to replace it. The links below give step by step DIY instructions on troubleshooting and attempting to fix this problem. You can disregard references to sand blockages as the cause. An older Canon camera is used to demonstrate these procedures but, the lens structure of these compact cameras is the same so the examples do apply to this unit. These procedures are effective about 60% of the time. Click on or copy the link location and paste them into your browser.
Please keep in mind that my diagnosis & solutions provided are directly
dependent on the accuracy of your description of the problem. As with
any "do it yourself" fixes, success is a "team effort", since I can't
see or touch the camera, and relies on the customer's manual dexterity
and ability to follow the instructions well.
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