When appending text, use following syntax
update
tablename
set
fieldname = convert(nvarchar(max),fieldname) + 'appended string'
where
fieldname2 = value2
or if you have old text and newly added text in one string. ( also you better use parameters)
cmd.CommandText = "update Client set Notes = ? where [Membership Number]= ?"
cmd.Parameters.AddWithValue("@param1", combinedText)
cmd.Parameters.AddWithValue("@param2", Membership_NumberTextBox.Text)