That usually isn't the only information you have.
For example, when I searched for HT-2021-000213 on Google I found this law360 article, which I guess might be the same as the one you have read. It's behind a paywall, but I can access enough of the article to see this text:
The case is Twitch Interactive Inc. v. Viewers Labs Ltd. and others,
case number HT-2021-000213, in the High Court of Justice of England
Using the party names (e.g. Twitch Interactive v Viewers Labs), it is possible to search most reported cases using a legal database such as Westlaw or Lexis (note that these are subscription based services). Many cases (including many unreported cases) also appear on Bailii, which is free.
You can also call the court, give the case reference number, and ask for more information e.g. the party names, whether the case has been decided, the outcome, a copy of the transcript (probably for a fee).