I would have thought not. The actual powers of the chairman are very limited, the main one being the power to call council meetings. The clerk should prepare agendas, and although these should be discussed with the chairman so as to ensure that business is dealt with effectively, the chairman has no power over the agenda.
In principle, the clerk should avoid favouring some councillors over others, and treat all equally, including the chairman and vice chairman. Of course, this may not always happen, and it is difficult to legislate for all information that might become available. But the council is free to decide rules about, for example, whether all letters to the council should be noted and reported to the whole council.