With this last explanation you have made this present discussion perfect. In a verb phrase, the auxiliary verb 'have', just like the auxiliary verb 'do', can't be used, that is to say never used, in a past-participle form, or perhaps in any non-finite forms - infinitive and participles. Thus in my mistaken verb phrase - 'have had gone', the second auxiliary verb 'have' is in either a finite past tense form (finite forms used only for leftmost verb) or a non-finite past participle form (you have just ruled out) and such construct is never to be used as a rule.

As long as learners understand the correct meaning of any construct, they tend to think it correct till got corrected using grammar!