In a word, no. To quote the Good Councillor's Guide:-
"The chairman has few special powers. For instance, it is
unlawful for a council to delegate decision making to any
individual councillor and the chairman is no different."
Similarly, the Clerk has a duty to bring important correspondence to the attention of the whole council. The Clerk is the proper officer of the whole council, not the PA of the Chairman.
Members should ask the Chairman and the Clerk to explain their actions.
The Chairman, as a local resident, may engage in whatever correspondence he or she wishes in a personal capacity, but in so doing, may not use their status as an elected representative of the people.