Critical message: Unable to access or write to cache directory! This warnings appear in your admin messages when AbanteCart's System Check is Enabled in Admin > System > Settings > System.
Each file and directory on your server is assigned access rights for the owner of the file, the members of a group of related users, and everybody else. Rights can be assigned to read a file, to write a file, and to execute a file (i.e., run the file as a program). This is important because AbanteCart may need access to write to files to enable certain functions or disable access to write for the public users (world).
/system/cache/ directory should be set with 777 permissions.
To change permissions use your hosting cPanel file-manager or an FTP Client. FTP programs ("clients") allow you to set permissions for files and directories on your remote host. This function is often called chmod (chmod is a UNIX command that means "change mode" on a file.) or set permissions in the program menu. Right-click 'file' and select 'File Permissions'
Use FTP client to change permissions or contact your host to solve this.