The only reliable way to prove what you have sent is to include a logging BCC address in the message. If that logging account is managed by the IT department, you can ask them to extract a copy of the message, that will be timestamped by the mail server. This along with the mail server logs can give an evidence of what was sent, at what time and what were the recipients.
In the absence of authentication on the mail server, that would only prove that your mail address was used, if the mail server requires an authentication, this would also prove that your account was used, but only a smart card signature would prove that you have sent that.
Ok this is the ideal use case, unfortunately, such logging BCC are uncommon, but it is easy to mimic them by adding your manager as a BCC to all important mails that you send. I will assume now that nothing of that have been used here.
As you can easily modify your local mailboxes, you cannot really rely on them to prove what you have done. At best, you can show their content, but your manager will be free to trust you or not. Things can be better if your mailboxes (including the sent messages one) are hosted in the mail server (can be common in IMAP mode). If you can only use SMTP or IMAP to access them, then you cannot easily tamper a message, or at least the server logs could have kept traces. In that case, the IT department could tell your manager whether according to their logs and knowledge it is likely that you could have changed the message or not.
I must acknowledge that this is not a very precise answer. At least it gives some hints on some places you could search for evidences. But if your sent messages are only kept on your local machine, unless you can prove that you have neither the tools nor the knowledge to modify the local mailboxes it will be very hard to prove anything, except by asking the client to send back to your manager the mail you had sent them. Unfortunately it may or not be possible depending on the client policy rules of your organization.