All correspondence received by the Clerk is actually to the Council. Hence, all members of the Council have an a right to view all or any of it. The clerk has no right to prevent this,
However, how the council actually deals with correspondence in practical terms is another matter. It is for the council to decide if it wishes all correspondence to be read at meetings or, for example, only items that the clerk considers necessary with the remainder simply being mentioned.
The obvious answer in the absence of an agreed policy would be for the clerk to introduce each item of correspondence and read out only those items requestedby members.