The relevant fragments of Hegesippus can be found in section 4:22 of Eusebius' Church History. We read there:
"And the church of the Corinthians continued in the orthodox faith up to the time when Primus was bishop in Corinth. I had some intercourse with these brethren on my voyage to Rome, when I spent several days with the Corinthians, during which we were mutually refreshed by the orthodox faith. On my arrival at Rome, I drew up a list of the succession of bishops down to Anicetus, whose deacon was Eleutherus. To Anicetus succeeded Soter, and after him came Eleutherus. But in the case of every succession, and in every city, the state of affairs is in accordance with the teaching of the Law and of the Prophets and of the Lord."
No knowledgeable Evangelical would deny that the history of the episcopate in the ancient churches, particularly apostolic churches like Corinth and Rome, is of interest in multiple contexts. Those bishop lists have historical value, and the continuity of teaching among the bishops is of value in arguing against heretics like those Hegesippus was opposing. Such a use of apostolic succession, if we're to call it that, doesn't carry with it the implications Catholics assign to their notion of apostolic succession in disputes with Evangelicals. Nothing Hegesippus says implies that bishops have inherited the office of the apostles, that every Christian church throughout history, including ones in circumstances much different from those of Hegesippus, must have a line of bishops going back to the apostles, etc. There isn't much in Hegesippus for a Catholic to use against Evangelicalism.
But we gain some information about Hegesippus from a later source, Epiphanius, that's problematic for Catholicism. Robert Lee Williams explains:
In the early twentieth century Lawlor compiled evidence that Epiphanius had preserved some information from Hegesippus that is not included by Eusebius but dovetails with Hegesippan material he has used....Furthermore, the Roman bishop list extends precisely to Anicetus in Epiphanius (Pan. 27.6), as Hegesippus states of his own according to Eusebius (Hist. eccl. 4.22.3)....
B.H. Streeter has given substantial evidence that both Irenaeus and Epiphanius derived independently from Hegesippus their comments on Marcellina and the Carpocratians and their Roman bishop lists. Irenaeus says of Marcellina that she "came to Rome under Anicetus" (Haer. 1.25.6). Epiphanius records of the same instance that she came "to us" (Pan. 27.6), but he was of Cyprus, not Rome. A common source appears likely. A common source is suggested again when both Irenaeus's bishop list and Epiphanius refer to Pauline epistles (Haer. 3.3.3; Pan. 27.6). Hegesippus is the only earlier writer known who could be this source....
In Rome the apostles Peter and Paul were the first bishops (Epiphanius, Pan. 27.6)....
Hegesippus was probably a source for Irenaeus's list of Roman bishops. Comparison of comments on Roman episcopal information suggest Hegesippus's doing his own research and Irenaeus's use of something already available. (Bishop Lists [Piscataway, New Jersey: Gorgias Press, 2005], pp. 96-97, 112, 129)
I'm currently focused on Hegesippus, but his view of Peter and Paul has implications for other sources, like Irenaeus. I'll have more to say about that when I address Irenaeus later.
For now, note that Epiphanius, apparently using Hegesippus' material, assigns a joint episcopacy to Peter and Paul. He doesn't say that they served as bishop of Rome at different times. Rather, he says that Peter and Paul filled one Roman episcopate at the same time, followed by Linus. Such a joint episcopacy can be reconciled with a Petrine papacy. But it's more naturally taken as evidence that the two apostles were viewed as equals. Peter and Paul are frequently grouped together in the earliest patristic generations, including in Roman sources. It isn't until the third century that we see a much more consistent and widespread singling out of Peter and various concepts of Petrine primacy.