In general, you cannot.
Particularly when it comes to media files (like music or video), if a web page changes its contents, whatever was there before is removed from public access.
However, there are a couple of things to try.
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The Internet Archive at archive.org has something they call the “Wayback Machine”, which lets you access archived copies of web sites from the past.
The problem with the Wayback Machine, however, is that not all sites are archived, and not all archives are updated frequently. Depending on the site you want to look at, you may or may not find a copy of the site as of the date you care to see.
An additional problem is that quite often, media – audios, videos, images, and the like – are not archived, for a variety of reasons.
And, of course, website owners can “opt out”, meaning that they instruct archive.org to never archive their site.
As an example, here’s Ask Leo! as of October 2003, courtesy of the WayBack Machine.
Most, but not all, of the links on the archived page even work.
Archive.org might be a place for you to start.
Many search engines – Google being the most notable – make available a “cached” copy of web pages they return in search results.
The problem here, however, is that cached copies are usually only the copy as of the most recent visit by the search engine indexer. Depending on the site, that could be anything as recent as a few hours, a few weeks, or more. If they haven’t updated their cache since whatever you were looking for was there, you might be able to find it. As an example, a cached copy of the Ask Leo! home page was one day old.
But once again, the cache focuses on web pages, typically does not include media such as audio or video, and can be discouraged by the web site owners.
The web is both permanent and ephemeral
We often say that once something is posted on the internet, it’s there forever. This certainly seems to apply to embarrassing photos, misstatements of truth, captioned cat pictures, and pretty much anything else that we might wish would go away.
On the other hand, it seems that situations such as yours are common as well: things we remember finding on the web disappear.
Either way, what remains on the web – permanently or briefly – remains out of our control.