Is there a way to hook into the update-core page for custom messages?

Question

We have some sites where we note a few things that need to be checked or done either before or immediately after a core, theme or plugin upgrade. While I keep a list of these things, sometimes users themselves do the upgrade and mess everything up.

I was hoping to be able to add a message to the upgrade screen to remind myself of the checklist of to do's before/after upgrade and also to warn users to contact me etc.

I'm thinking if there is a hook to let me add this customized message that'd be great. If not any suggestions how to approach this? Thanks!