Remove the batteries, 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. If you have been using the same rechargeable battery life of the camera (more than 3-5 years) then the power capacity of the batteries may not be high enough anymore to handle the increased load or the battery does not really charge deeply anymore, meaning the power dips immediately under load. Because age is the big factor in battery health, even if they were equally alternated in use, a new batteries may be the answer. In addition, It is a known issue that digital cameras that use only 2 AA size batteries, tend to require a little more power as they age due to deteriorating parts and surface residue. Lens movement uses the most power. That is why the connection is so critical. The only other power suggestion that I have is to try a set of high power non- rechargeable lithium AA batteries.
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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Also, try powering the camera on with the memory card removed. If it powers on without the card installed, please let me know.