Can you get an old news story about you removed?
Rarely removed, and frequently made much harder to find — which is the practical distinction, because the two routes available do completely different things and people usually want the second while asking for the first.
The two separate routes:
Asking the publisher to remove or amend the article. Publishers are extremely reluctant, because the archive is a historical record and altering it undermines its integrity. They will correct inaccuracy, and they will rarely delete an accurate report of something that happened.
What publishers sometimes do instead: add an update noting a later outcome — an acquittal, a dropped charge, a spent conviction, a correction; remove the article from search engine indexing while keeping it on the site; or, occasionally, anonymise.
Asking a search engine to de-list it. Under data protection law, individuals can request that results for searches of their name be removed where the information is inadequate, irrelevant or excessive — the so-called right to be forgotten.
What de-listing actually achieves: the article remains published; it simply no longer appears when someone searches your name. For most people this is what they actually want, since the harm is being found rather than the existence of the record.
How the balance is struck. The request is weighed against the public interest in access — considering the seriousness of the matter, how long ago it was, whether you hold public office or a position of responsibility, whether the information relates to criminal offences and whether they are spent, and whether you were a child at the time.
A person with a public role is far less likely to succeed, which is the deliberate design.
What tends to succeed: old, minor matters concerning private individuals; matters where the outcome was favourable and the article does not say so; inaccuracies; and content concerning someone who was a minor.
What tends not to: recent matters, serious offences, professional misconduct, and anything concerning a public figure acting in that capacity.
Where to escalate: the search engine's own process first, then the data protection regulator. For the article itself, the publisher's complaints process, then the relevant press regulator.
Court reporting has its own restrictions, and some anonymity is automatic and permanent.
General information, not legal advice.