my $feedurl = $ua->get($url)->result->json->{feeds.meta.links.feed}; which gave error message: Bareword "feeds" not allowed while "strict subs" in use

Only very simple barewords are automatically quoted in $hash{...} keys, the . is seen as an operator here. Try ...->json->{'feeds.meta.links.feed'};.

Update: See Re: hash key.

Lets take it back a step, there's various ways to achieve what you want. Perhaps this example is clearer:

#!/usr/bin/perl

use strict;
use warnings;
use Mojo::UserAgent;

my $url = 'https://public-api.wordpress.com/rest/v1/read/feed/?url=the
+-art-of-autism.com';

my $ua = Mojo::UserAgent->new;

my $json = $ua->get( $url )->res->json; 
for my $feed ( @{$json->{feeds}} ){
  if ( $feed->{meta}{links}{feed} ){
    print "$feed->{meta}{links}{feed}\n";
  }
}
[download]

For each entry in feeds, print the value of meta->links->feed.

Update: see Re^4: Running JavaScript from within Perl (or just use the API).