Here's "have gone" and "have went" discussed on Linguist List, where the conclusion is that neither is exactly incorrect:
With that in mind, if you belong to a dialect community in which people consistently say "I have went..." instead of "I have gone...", then among your friends and community, there is nothing wrong with "It seems to have went well." If writing to someone outside the community, or a formal document or school assignment, it were better to use "It seems to have gone well." If your community of English generally says "I have gone..", then "to have went..." in that case is in fact "incorrect", that is, ungrammatical -- contrary to the patterns of that dialect.
This discussion about "would have went" also suggests that speakers who do use the phrase might switch to "would have gone" in a setting that demands a formal register, like the classroom or in writing.
Also discussing the need to switch dialects is this paper, Responding to African American Vernacular English (AAVE) in Written Assignments [PDF, see page 13] - the author suggests that "have went" is common in spoken AAVE but may not be common in writing even when other features of AAVE are present:
Constructions as that in the first sentence [I had did report cards for my student teaching...], although
common in AAVE (the use of the past tense rather than
past participle in irregular verbs: “I should have went”)
are very rare in these papers. Almost all AAVE usages
involve dropped endings.
(The writer is discussing a small sample of graduate student work selected for analysis)
I don't know enough to explain exactly why some dialects of English use the past participle and some use the past tense, or whether the use is appreciably changing in recent years.
I've found plenty of half-formed theories for why past tense is used, but the only idea I see that doesn't totally rely on putting down speakers of one dialect or another is that "gone" is 'a past participle which is not formed by the simple addition of -ED' and so perhaps a regularization is occurring. This idea is a little more obvious if you look at some examples of regular and irregular past tense and past participles:
present | past | pp