underwent is the simple past tense. undergone is the past participle (so it is used in compound tenses). It is like go.
go went gone
undergo underwent undergone.
Underwent is not paired with "have." However, undergone is paired with "have."
I undergo a difficult commute every day.
I underwent a particularly difficult one yesterday.
I have undergone much more difficult ones in the past.
I go to the store on Wednesday.
I went to the store on Monday this week.
I have gone to the store on other days of the week.
Remember, the third person changes the form:
She goes to work.
She went to work
She has gone to work every day since she started.
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