You miss a shortcut file for your program in $HOME/.PlayOnLinux/shortcuts/ directory. That's the directory which PlayOnLinux uses to read shortcut files you see in the PlayOnLinux program window.
Make sure you have the shortcut. Please keep in mind, that many PlayOnLinux scripts may be outdated (using very old Wine version with tricks which are not needed in newer Wine versions). Personally, I always prefer "install a non-listed program" method, and if needed, I check Wine Application Database (Wine AppDB) for further guidance (for example, which DLL overrides/libraries are required to run the program etc).
You find League of Legends AppDB site here: https://appdb.winehq.org/objectManager.php?sClass=version&iId=33303
This is what you have to do (basically):
1. On PlayOnLinux main window, open Tools -> Manage Wine Versions
2. Install Wine 2.4 or Wine 2.5 (x86 is the right choice I guess)
3. After Wine has been installed, open PlayOnLinux main window and click "install a program". Click "install a non-listed program".
4. Choose "Install a program in a new virtual drive". Give it the name whatever you ever want.
5. Check "Use another version of Wine".
6. On the next windows, select Wine 2.4/2.5 and 32-bit windows installation. Then, install the game (mount CD/DVD on your Linux system and find setup executable if necessary).
7. After the installation, create a new shortcut for the game (PlayOnLinux asks you about that).
8. Right click the app shortcut on PlayOnLinux main window. Select "Configure Wine", open "Libraries" tab and set new override for msvcp140 (select "Native then Builtin"). Close the Wine configuration window.
According to the instructions on Wine AppDB site, the game should work with these settings. For further guidance, please see the AppDB page again (link above).
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P.S. PlayOnLinux uses its own POL "Install components" method which you can consider quite same than what winetricks do. "Install components" is not exactly the same thing than winetricks but you can install those components as they're possible been referred on Wine AppDB program specific site.
You can install these extra components during the program installation (install a non-listed program, League in this case) and by checking "Install some libraries" during the installation process. You can also do that process afterwards by selecting the program shortcut on PlayOnLinux main window and clicking "Configure" button in the top bar (right side).
