My Fujifilm AX560 will not stay turned off

It takes 4 x AA batteries. Are the battery contacts on the compartment door, and the battery contacts in the base of the compartment clean and shiny. Usually when batteries leak, they cause oxidization on the metal.

The camera does have very specific requirements.

Batteries are required to operate the digital camera. At
first, load the batteries in the camera.
Compatible batteries
AA-size alkaline batteries (4) or AA-size Ni-MH
rechargeable batteries (4) (sold separately)
CAUTION
• Never use the following types of
batteries, as they could cause
serious problems such as battery
fluid leakage or overheating:
1. Batteries with split or peeling
outer casings
2. Combinations of different
types of batteries or new and used batteries together
• Do not use manganese, Ni-Cd or lithium batteries.
• Use the same brand and grade AA-size alkaline
batteries as those supplied with the camera.

https://www.manualslib.com/manual/54755/Fujifilm-Finepix-S8000fd.html?page=16#manual

And on the next page it shows the polarity two up, and two down. If those weren't correct, you would be on reduced power.

..

check the mhr on the batteries you use
they should be re-chargeable batteries as alkaline batteries do not have what it takes and run down very quickly
the re-chargeable battery should be around the 2000 mhr rating

Did you chaco if the contacts in the battery holder are clean and shiny?
Doy you have some rechargeable NiMH batteries. Try them, because they have more capacity then the alkaline ones.
Never use AA manganese (Zinc-Carbon) batteries or NiCd rechargeable, because the voltage is too low.

That's easy. It never charges.
The Fujifilm AX560 is a camera, not a battery charger. If you want to use rechargeable batteries then you must charge them outside the camera using a separate charger.

Hit menu, left and down to Set, then right and down many times to the second last item on screen 5 to get to battery type. Supports Lithium, Alkaline, Ni-MH

Your camera takes two AA sized batteries. You can use either disposable alkaline cells (like Duracell) but the battery life will be short, or can use rechargeable NiMH cells (recommended). Absolutely do not use NiCad or Lithium rechargeables in your camera, and you also mustn't use manganese cells (sometimes called lithium manganese).

Please take a moment to rate my answer.

You didn't mention what of battery you used.

If you using the the rechargeable you must use the Ni-MH batteries or lithium but i suggest to used the Ni-MH

If non rechargeable you can use alkaline.

Now If your using the alkaline and camera will not stay On try to reverse the battery orientation instead of negative positive, make it positive negative. If ths camera will not turn On it means camera needs of service you may call tech support toll free 1 800 235 6325 ext 12

try new Alkaline batteries or fully charged NiMH batteries, you may need to clean and adjust the springs in the battery compartment
this tips may give you more details
http://www.myfixtips.com/blog/digital-camera/why-my-camera-keeps-saying-%e2%80%98change-battery%e2%80%99-even-with-new-batteries-176.htm

There are three battery type settings in the set up menu: Alkaline, COOLPiX(NiMH) and Lithium. Might be worth looking at. To be continued.

use batteries with lithium mine has lasted me over a year