Just tell them what you have told us.
It is just a simple email, not the great Indian novel. Don't overthink these things.
You might need to rethink the process. How do they communicate the manager's approval? Is it a form that has to be filled in, and signed by manager? Then do they need to scan that and attach to email. Get your process sorted out before you email the staff - and keep it simple. Staff won't comply if the process is onerous.
Second point, why do staff need to be involved? Why can't the manager send the approval to HR?