I guess apple devices make those requests if the device owner adds the site to it. This is the equivalent of the favicon. To resolve, add 2 100×100 png files, save it as apple-touch-icon-precomposed.png and apple-touch-icon.png and upload it to the root directory of the server. After that, the error should be gone.
I noticed lots of requests for apple-touch-icon-precomposed.png and apple-touch-icon.png in the logs that tried to load the images from the root directory of the site. I first thought it was a misconfiguration of the mobile theme and plugin, but found out later that Apple devices make those requests if the device owner adds the site to it.
Source: Why Webmasters Should Analyze Their 404 Error Log (Mar 2012; by Martin Brinkmann)
If a user from Safari Web browser (Apple devices) visit your site. The browser tries to fetch the site icon if it is not defined in
in the following order:apple-touch-icon-57×57-precomposed.png
apple-touch-icon-57×57.png
apple-touch-icon-precomposed.png
apple-touch-icon.png
To resolve this issue either define an icon for safari web browsers or apple devices. Add something like this to head section of your site:
If you want to keep
clean then upload the icon to root dir of your site with proper name.The default icon size is 57px.
You can find more details on iOS developer library.