I also have many Peckhams, including Johns from Wadhurst and Mayfield around 1750-1850, but do not find a John b.Wadhurst 1823 in my tree - do you have any more info on him e.g. wife, children, death/marriage data ?
Maybe might be able to guess a potential link through childrens names, or at least eliminate other various John Peckhams that I have traced back.