Thanks to all who offered advice (upvotes all round). I took bits of information from various responses; this is a general reply to everyone. I determined a workaround, which I'll document below for anyone encountering the same problem. I suspect an actual solution would be to rebuild Perl and reinstall all modules (a fairly lengthy task which I really don't have time for right now).

Information regarding the missing xlocale.h: "glibc wiki: Removal of 'xlocale.h'".
Information about how this affects Perl: "glibc-2.26 removed xlocale.h".
Based on that information, I edited config.h (which is in the same directory as perl.h in the error message).

$ ls -al config.h*
-r--r--r-- 1 ken None 164422 Jun 23  2020 config.h
$ cp config.h config.h-20210102a
$ chmod +w config.h
$ vi config.h
$ chmod -w config.h
$ ls -al config.h*
-r--r--r-- 1 ken None 164425 Jan  2 10:55 config.h
-r--r--r-- 1 ken None 164422 Jan  2 10:51 config.h-20210102a
$ diff config.h config.h-20210102a
3096c3096
< /*#define     I_XLOCALE               / **/
---
> #define       I_XLOCALE               /**/
[download]

Now back to cpan:

cpan[1]> install XML::LibXML
...
  SHLOMIF/XML-LibXML-2.0206.tar.gz
  /usr/bin/make install  -- OK

cpan[2]>
[download]

And a very quick confirmation:

$ perl -E 'use XML::LibXML; say $XML::LibXML::VERSION'
2.0206
[download]