If the minutes were not correct, they should have been amended before being signed as the permanent record of the proceedings. There is no process by which the Chairman may add a confidential addendum to the minutes. If the Chairman believes that the minutes are incorrect, but the other members do not agree, there is a process set out in standing orders for the Chairman to sign the minutes, but record the reason for disputing their accuracy.
Which committee held the initial discussion? You should have a policy for dealing with staffing matters of this type and that policy must be followed at all times. A good policy will clarify the roles of individuals, committees and the Council at every stage of the process.
As an aside, when discussing confidential matters, it is important that the procedure for excluding the press and public is followed, even if nobody is present. If you haven't passed a resolution to exclude and somebody walks in, you have no right to ask them to leave. Technically, anything discussed at a meeting without passing an exclusion motion should be included in the public minutes of the meeting.