I don't understand what you're trying to do. Note that according to perlrun, the -a autosplit switch can only be used with the -n or -p switches. (Update: The fact that you have what looks like a file name at the end of the OPed pseudo-code suggests you intended to use a -n or -p switch.)

You seem to be trying to use the ! $seen{ $string }++ idiom. Maybe the following will help clarify the situation although it doesn't actually use the autosplit feature (but neither does the OPed (pseudo)code).

Win8 Strawberry 5.30.3.1 (64)  Sun 07/17/2022 22:13:06
C:\@Work\Perl\monks\abdan
>cat story.rst
seen first
the first found
found second
seen second
first occurred
occurred second

>perl -ane "print if /(seen|found|occurred)/ && !$seen{ $1 }++" story.
+rst
seen first
the first found
first occurred
[download]

It is uncool to update a node in a way that renders replies confusing or meaningless.

this code doesn't compile or even run

What hippo said++.

The /$_(\h\.)? fragment still does not compile. If I "fix" it to /$_(\h\.)?/ it doesn't make sense because $_ is the line/record read from the input file (story.rst in the example), so the equivalent expression is
    $_ =~ /$_(\h\.)?/
The (\h\.)? tacked on at the end is optional and therefore kinda meaningless, so the expression more or less (!) reduces to
    $_ eq $_
unless $_ contains regex metacharacters, in which case all bets are off.

A possible way forward would be for you to post a small example of the typical content of the story.rst file and of the output you expect from processing the given content (update: see Short, Self-Contained, Correct Example for this general technique). The best way to do this is usually as a reply to your OP. If you want to update the OP, please see How do I change/delete my post? as suggested by hippo.

... is astray to s.t. ...

I still don't understand what this means.

N.B: The statement
    print if $seen{ $string }++
will print every line after the line in which $string (however $string is derived) is first found. Is this what you intend?

print if /...(...).../ && $seen{$1}++;
[download]

(The pattern you had made no sense.)

The above prints the duplicates. Don't forget to negate if you are trying to remove duplicates.

print if /...(...).../ && !$seen{$1}++;
[download]