This is always tricky as people tend to think of email as private; but as John1706 says, all council email accounts are potentially subject to FOI requests. The problem arises when councillors need to discuss anything confidential, such as complaints, or employment related matters - this would not be released under FOI, and the Clerk should not automatically have access either.
I would suggest making it clear to the clerk that he/she must have permission from the Chairman/relevant committee to read councillor emails, and must give a specific reason, time interval and relevant email accounts each time.
And then ensure all councillors are aware that anything sent from council email accounts is potentially public communication. It is then up to them to find other ways for private / non-council business communication - and even then it is incredibly easy to accidentally send an email to the wrong person.
It may also be worth considering what happens to any councillors', and the clerk's, email accounts when someone resigns. The new clerk would likely take over the account from the previous clerk, and it is likely that ex-councillor email accounts would be kept accessible for the Clerk/Chairman for some time, in case there are any ongoing issues where someone needs to refer back to emails on that account