An eBay community member has suggested a possible workaround for this issue would be to reply to messages directly within your email client instead of by clicking the blue "reply" button.
I also am currently getting an error message when sending messages using the eBay app or a mobile browser.
However, I am able to send messages using Firefox or Edge on a desktop computer (Windows 10).
If you have eBay messages sent to your email, you may be able to reply to the email directly (rather than using the blue "reply" button in the message) to send an eBay message to the user.
eBay users are encountering a failure error when trying to send messages through both desktop website and mobile app.
Fail to add member message
The code Trading.20965 seems to indicate the problem is in the eBay Trading API and in fact that code is listed as a "serious error" in eBay's developer documentation.
Unfortunately, that means there's nothing affected users can do except wait for eBay to fix whatever is causing the error on their side and try to send the message again at a later time - a frustrating experience to be sure for users who are trying to complete sales or ask pre or post purchase questions.