Please be aware that resolving this issue can take multiple attempts. With some units that have had this problem it has taken me up to 6 or 7 hours of manipulation, over a few days, to make the lens move freely.
My first inclination is to assume that a lens error is the problem itself, but first, there is one more likely possibility, lack of sufficient power. Remove the battery, check the pins or connectors in the battery compartment, and make sure that they are positioned correctly to contact the batteries. Then use a q-tip with alcohol to clean and dry them. It is a known issue that as digital cameras age, they tend to require a little more power as they age due to deteriorating parts and surface residue. Lens movement uses the most power. If no improvement, please continue.
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, if uneven, 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.
Even though the lens extends, it is quite possible that it does not make it to the correct sensor point due to an obstruction like a grain of sand or residue build up. A lens position error will halt the camera start up process. Lens errors are a common problem. Dirt can get into the gears or the lens can become misaligned from being carried in pockets, purses or at the beach. 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. The links below give step by step DIY instructions on troubleshooting and attempting to fix this problem. A 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 it into your browser.
LENS FIX 1
LENS FIX 2
If the "do it yourself" approach doesn't work out for you and you need conventional repair No local shops in your area make these kinds of repairs "on-site", they will send your camera to a regional repair service.
Note: While it is possible to have your camera cleaned and repaired, it is most likely that it will not be very cost effective for you. Considering the age and present value of this camera, paying the cost of having it repaired , approximately $97 - $110 really depends on how attached to the camera you are. The standard "rule of thumb" is if the repair cost is greater than 50% of the value of the camera, it is not worth it. A good used or refurbished DSC-W390 can be purchased for as low as $20 - $45 and I've seen a brand new one for around $65.
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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