Scenario
You may want to do some changes to your Jira and temporarily stop JEMHC reacting or feeding it.
Process
Inbound
You can go to JEMHC > Messaging > MessageSources and edit each to disable it (stopping inbound retrieval). Reverse process to re-enable.
Outbound
You can go to JEMHC > Messaging > Message Outbounds and edit each to disable all mail sending (but events will still be captured, see next). Reverse process to re-enable.
Webhook Event Capture (that drive notifications)
You can go to JEMHC > Notifications > Email and edit each to disable the Notification Mapping, this prevents JEMHC storing changes for the included Projects(s). Reverse process to re-enable.