コード

<?php
$results = [
    ['name' => '田中',  'japanese' => 83, 'math' => 57, 'science' => 43,  'social' => 72, 'english' => 78],
    ['name' => '渡部',  'japanese' => 62, 'math' => 88, 'science' => 70,  'social' => 66, 'english' => 38],
    ['name' => '品川',  'japanese' => 23, 'math' => 33, 'science' => 53,  'social' => 17, 'english' => 7],
    ['name' => '滝谷',  'japanese' => 93, 'math' => 89, 'science' => 79,  'social' => 96, 'english' => 91],
    ['name' => '川原田','japanese' => 55, 'math' => 56, 'science' => 60,  'social' => 54, 'english' => 66],
    ['name' => '一色',  'japanese' => 27, 'math' => 47, 'science' => 100, 'social' => 42, 'english' => 38],
];
$new_results = [];
foreach($results as $key =>$value){
    $new_results[$key] = $value;
    $new_results[$key]['gokei'] = $value['japanese'] + $value['math'] + $value['science'] + $value['english'];
}
var_dump($new_results);

結果

array(6) {
  [0]=>
  array(7) {
    ["name"]=>
    string(6) "田中"
    ["japanese"]=>
    int(83)
    ["math"]=>
    int(57)
    ["science"]=>
    int(43)
    ["social"]=>
    int(72)
    ["english"]=>
    int(78)
    ["gokei"]=>
    int(261)
  }
  [1]=>
  array(7) {
    ["name"]=>
    string(6) "渡部"
    ["japanese"]=>
    int(62)
    ["math"]=>
    int(88)
    ["science"]=>
    int(70)
    ["social"]=>
    int(66)
    ["english"]=>
    int(38)
    ["gokei"]=>
    int(258)
  }
}