Add a specific label like "clickup" or "sent-with-clickup" to every mail which was sent from the app.
Then you could setup a rule in Gmail to auto archive or at least auto move those mails from the inbox.
Actually you get the answer for an email sent with clickup inside the task AND in the gmail inbox what can be confusing.
I would love to archive them directly, knowing that I have them inside clickup with all needed notification.
I use the "Save Emails and Attachments" AddOn for Google Spreadsheets / Gmail to auto download recurring invoices sent via Mail. That little AddOn sets a label "saved" to those emails and with this I dont need to watch them again in my inbox. They go directly to archiv.
Love to work with clickUp