The cPanel MySQL update scripts can crash your MySQL server and it won't come back up. If this happens, you can remove the partial updates files and roll it back to the former version of MySQL. Doing an update after restoring the old service is sometimes completed successfully. (Worked for us during the MySQL 5.6 to MySQL5.7 crashed update).

STEP 1: Remove the mysql rpms 

sudo yum remove mysql*

STEP 2: Remove all mySQL repos from /etc/yum.repos.d

cd /etc/yum.repos.d

rm mysql*

STEP 3: Manually set mysql to 5.6 in /var/cpanel/cpanel.config

vi /var/cpanel/cpanel.config

STEP 4:Run cPanel script to intall missing rpms

/scripts/check_cpanel_rpms

STEP 5: in WHM, run the SQL Update again.

If getting "Unmanged" error on update, you need to set the cPanel rpm's to managed (this case for MySQL56)...

/scripts/update_local_rpm_versions --edit target_settings.MySQL56 installed

/scripts/update_local_rpm_versions --del target_settings.MySQL56

 Check the config file...

vi /var/cpanel/rpm.versions.d/local.versions

It should like like this...

---
file_format:
 version: 2
install_targets: {}
rpm_locations: {}
srpm_sub_packages: {}
srpm_versions: {}
target_settings:
easy-tomcat7: uninstalled
perl522: uninstalled 
perl524: uninstalled
perl526: uninstalled
url_templates: {}

 Other Reference...

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If Installing MYSQL from scratch... (not recommended - use cPanel)

Instructions:
https://dev.mysql.com/doc/refman/5.6/en/linux-installation-yum-repo.html

**Get 5.6 RPM
wget https://dev.mysql.com/get/mysql80-community-release-el6-3.noarch.rpm
sudo yum localinstall mysql80-community-release-el6-3.noarch.rpm

 yum repolist all | grep mysql

sudo yum-config-manager --disable mysql57-community
sudo yum-config-manager --disable mysql80-community
sudo yum-config-manager --enable mysql56-community
yum repolist enabled | grep mysql

sudo yum install mysql-community-server

sudo service mysqld start