That sounds like a resource issue. How much storage space do you have left? (You can check this by going into Settings|Storage|Internal (sometimes called Phone).
The fact that the icons reappear after a restart tends to comfirm this. Does clearing recent apps also bring them back?
I'd suggest deleting anything you don't need - that would include apps and content. I'd also recommend adding an sd card for content storage. You can set the camera to use it by default (once the card is inserted, you may find that openign the camera offers you the option immediately). You may also be able to set any music apps to download to it in the first instance. Content already stored can be moved by using the built-in file manager app, though I'd recommend copying first and only deleting once you've confirmed that the files are safely in the new location.
Photos can also be backed up to cloud storage - Google Photos is good for this as its "high resolution" (actually slightly compressed) option doesn't count against your storage allowance. The process is automatic and can be set to run over wifi only to preserve your mobile data.
It's also possible to move some apps to a card, although that's often less than satisfactory. Simply using the card for content should be enough.
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