I can't understand where the problem is. Apache is running with my standard user that actually can write those files.
What should I do?
Check SElinux is enabled or not using the following command from su in console.
if it returns "enabled" then disable it using following command
now check that apache can write to the user directory.
One possibility is that you have SELinux enabled and the files/directories that can't be written were originally created in your home directory (e.g. you unzipped them there, then moved then to /var/www/html). SELinux doesn't allow apache to modify those files even though they seem to have correct file permissions. Try disabling SELinux. If that resolves the problem, you can either run with SELinux permanently disabled, or you can work through the SELinux settings to get those files and directories writable by apache.