for (values %$HASH1) {
    my %seen;
    $_ = [ grep { ! $seen{$_}++ } @$_ ];
}

##</code><code>##

my %seen;
for (values %$HASH1) {
    $_ = [ grep { ! $seen{$_}++ } @$_ ];
}

##</code><code>##

my %seen;
$_ = [ grep { ! $seen{$_}++ } @$_ ] for values %$HASH1;

##</code><code>##

{
  "Alabama"    => ["Andalusia", "Anniston", "Clanton", "Eufaula", "Auburn"],
  "California" => ["Barstow"],
  "New York"   => ["Amsterdam", "Coney Island", "Beacon"],
  "Utah"       => ["Layton"],
}

##</code><code>##

$_ = [ grep { state %seen; ! $seen{$_}++ } @$_ ] for values %$HASH1;