You could solve both problems in one go by temporarily inverting your hash. So instead of a (state => cities) hash, you have a (city => state) hash. Then you can invert it back.

use strict;
use warnings;
use Data::Dumper;

my %hash = (
  'Alabama'    => ['Andalusia','Anniston','Clanton','Eufaula','Auburn'
+,'Auburn','Auburn','Auburn'],  
  'California' => ['Barstow','Eufaula'],
  'New York'   => ['Amsterdam','Coney Island','Beacon','Beacon','Beaco
+n','Beacon','Beacon'],
  'Utah'       => ['Beacon','Layton'],
);

# Invert the hash.
my %inverted = map {
    my $state = $_;
    map { $_ => $state } @{$hash{$state}}
} sort keys %hash;

print Dumper \%inverted;

# And back again:
%hash = ();
while (my ($city, $state) = each %inverted) {
    push @{ $hash{$state} ||= [] }, $city;
}

print Dumper \%hash;
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Can i delete this for one key only

Of course. Simply amend your algorithm to maintain a separate list (which could be a hash for easy/fast lookups) of deleted values. Never delete a value which is already in that list.

"Can i delete this for one key only (no particular where i need to retain this value for which key, just need once in any key)"

I made reference to that in my original reply:

"To keep the values unique for a specific key, without removing duplicates from the values of other keys, you'll need to clear the %seen hash for each arrayref value."

In this code:

for (values %$HASH1) {
    my %seen;
    $_ = [ grep { ! $seen{$_}++ } @$_ ];
}
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Just move the "my %seen;" line out of the loop:

my %seen;
for (values %$HASH1) {
    $_ = [ grep { ! $seen{$_}++ } @$_ ];
}
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Which, as the loop now only contains one statement, you can write more succinctly as:

my %seen;
$_ = [ grep { ! $seen{$_}++ } @$_ ] for values %$HASH1;
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I put that into the original code I gave you (along with your new HoA). Here's the output:

{
  "Alabama"    => ["Andalusia", "Anniston", "Clanton", "Eufaula", "Aub
+urn"],
  "California" => ["Barstow"],
  "New York"   => ["Amsterdam", "Coney Island", "Beacon"],
  "Utah"       => ["Layton"],
}
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Which exactly matches what you have for "so my result hash should look like".

By the way, purely for academic interest, because I can't see that it gains you anything in this particular instance, if you're using Perl 5.10, or later, you could have reduced those two lines to this one:

$_ = [ grep { state %seen; ! $seen{$_}++ } @$_ ] for values %$HASH1;
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I tested that: the output is identical. See state for more details.