CloudLinux is a platform designed to improve the stability and security of shared hosting environments. It provides features like CageFS, LVE (Lightweight Virtual Environment), and others to isolate and limit the resources available to individual users on a server.

Regarding your specific question about web server and PHP handler support, CloudLinux is compatible with a wide range of web servers and PHP handlers. Here's a breakdown:

1. Nginx: Yes, CloudLinux supports Nginx. Many hosting providers use Nginx as a reverse proxy in combination with Apache to improve server performance.

2. PHP-FPM: Yes, CloudLinux supports PHP-FPM (FastCGI Process Manager). PHP-FPM is a highly efficient method for processing PHP scripts.

3. FCGID: Yes, CloudLinux supports FastCGI (fcgid). FastCGI is a popular protocol for interfacing applications with web servers.

4. LiteSpeed: Yes, CloudLinux supports LiteSpeed. LiteSpeed is a high-performance web server that is often used as an alternative to Apache.

5. Apache + mpm_itk: Yes, CloudLinux is compatible with Apache and mpm_itk. mpm_itk is a module for Apache that allows each VirtualHost to run as a separate user, which can be useful for shared hosting environments.

Keep in mind that software support and compatibility can evolve over time, so it's recommended to check with the official CloudLinux documentation or contact their support for the most up-to-date information if you're working with a version released after September 2021.