using a regular expression to search and remove the elements in place could be faster:

my $delete = join('|', map quotemeta, sort @element_to_delete);
my $re = qr/(?:^|,)(?:$delete)(?=,|$)/;

for (@key_to_search) {
  if (defined $new_reports{$_}) {
    $new_reports{$_} =~ s/$re//g;
    delete $new_reports{$_} unless length $new_reports{$_}
  }
}
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updated following ikegami advice