I was trying to apply Patch Version:  HF4 and the precheck was failing with errors:

TimezoneMisMatch – Timezone on the node is ABC Time and the required timezone is XYZ Time
ManagementAgent – ManagementAgent is not running, last update was 8 hours ago

Screenshots:

Seems like this is a known issue while applying the cumulative patch for vRA 7.6. In order to fix this issue, I had to perform following steps:

1. Set the timezone of vRA appliance to default UTC.
2. Set the timezone of all the IaaS machines to UTC (Reboot is required for IaaS Machines)
3. Run the precheck again from vRA VAMI.

Once the above steps are done, you would see the below:

Happy Patching!!