You said, ...push multiple key-value pairs onto an existing hash. Just to clarify, values (scalars) are pushed onto an array; you don't push onto a hash.

The push in your while loop adds array references as values associated with keys, so you see a mix of data from the Dumper output (your push @{$gene{$key}}, $value; notation says treat the key's associated value as an array and push the contents of $value onto that array):

$VAR1 = {
          'HES4' => '57801',
          'TMEM52' => [
                        '339456'
                      ],
          'SSU72' => [
                       '29101'
                     ],
          'GNB1' => [
                      '2782'
                    ],
          'PLEKHN1' => '84069',
          'NCRNA00115' => '79854',
          'ATAD3A' => [
                        '55210'
                      ],
          'ATAD3B' => [
                        '83858'
                      ],
          'SLC35E2' => [
                         '9906'
                       ],
          'SAMD11' => '148398',
          'NOC2L' => '26155'
        };
[download]

The [ ] notation indicates a list--each one having been generated by the push--so your hash now contains array references and non-reference values.

choroba has shown you how to add the key-value pairs to an existing hash (which will overwrite an existing value, if the key already exsits):

while(<DATA>){
    chomp;
    my ($key, $value) = split;
    $gene{$key} = $value; # choroba's suggestion
}
[download]

Dumper output:

$VAR1 = {
          'HES4' => '57801',
          'TMEM52' => '339456',
          'SSU72' => '29101',
          'GNB1' => '2782',
          'PLEKHN1' => '84069',
          'NCRNA00115' => '79854',
          'ATAD3A' => '55210',
          'ATAD3B' => '83858',
          'SLC35E2' => '9906',
          'SAMD11' => '148398',
          'NOC2L' => '26155'
        };
[download]

Is this what you wanted to achieve?